The concept of duality I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come.
Wikipedia has a good page on several forms of "duality" in mathematics, which outlines several notions of duality (geometric, in convex analysis, topology, set theory, etc.) I am very interested in getting help with the following goal:

Collect an annotated list of various notions of duality that occur in mathematics, with the ultimate aim of describing the notions in a way that makes it easier to recognize and intuitively build connections between the various notions of duality. Also welcome are comments / answers that highlight how a particular notion of duality can be extremely useful (in proving theorems, in applications, for computational reasons, etc.)

Some additional context
I got thinking about this question after reading the following amazing paper:
The concept of duality in convex analysis, and the characterization of the Legendre transform, by Shiri Artstein-Avidan and Vitali Milman, where the authors talk about duality in more abstract terms (though, largely in the setting of convex analysis). Motivated by their abstract treatment got me thinking whether such abstract treatments of duality have been investigated for other types of duality, which eventually led to this question.
Thus, in line with the Avidan-Milman results, one may also ask similar questions about other types of duality (i.e., one tries to characterize why and how a chosen notion of duality is the only "natural" choice under a set of axiomatic requirements).
 A: I enjoyed a series of talks by Bernd Sturmfels on some such interrelationships, which it looks like are written up in a paper by Rostalski and Sturmfels called "Dualities in Convex Algebraic Geometry."

Abstract: Convex algebraic geometry concerns the interplay between optimization
  theory and real algebraic geometry. Its objects of study include convex semialgebraic
  sets that arise in semideﬁnite programming and from sums of squares. This article
  compares three notions of duality that are relevant in these contexts: duality of convex
  bodies, duality of projective varieties, and the Karush-Kuhn-Tucker conditions derived
  from Lagrange duality. We show that the optimal value of a polynomial program is an
  algebraic function whose minimal polynomial is expressed by the hypersurface projectively dual to the constraint set. We give an exposition of recent results on the boundary
  structure of the convex hull of a compact variety, we contrast this to Lasserre’s representation as a spectrahedral shadow, and we explore the geometric underpinnings of
  semideﬁnite programming duality.

A: One "duality principle" that occurs in category theory is that of Isbel Duality. My feeling is (feel free to correct me if I am wrong) is that this encapsulates stone duality, Gelfand duality, and the duality of affine schemes and commutative rings in the same disscusion. Let $\mathcal{C}$ be a (small) category. Then the presheaves on $\mathcal{C}$ and the co-presheaves on $\mathcal{C}$ are somehow dual to one another. Conceptually, one thinks of the presheaves as spaces and co-presheaves as quantities.
This is something that I am trying to understand myself. Some nice articles at n-lab are:
https://ncatlab.org/nlab/show/space+and+quantity
https://ncatlab.org/nlab/show/Isbell%20duality
A: From Quantum Field Theory III: Gauge Theory by Eberhard Zeidler (Springer, 4/2011):
A) Preface (pg. XII):
"It turns out that cohomology and homology have their roots in the rules for electrical circuits formulated by Kirchhoff in 1847."
B) Ch. 22. Electrical Circuits as a Paradigm in Homology and Cohomology :
(pg. 1009)
"The study of electrical networks rests upon preliminary theory of graphs. … My purpose here is to show that actually this theory is nothing else than the first chapter of classical algebraic topology theory and may be very advantageously treated as such by the well known methods of that science." -- Solomon Lefschetz (as quoted by Zeidler) from Applications of Algebraic Topology: Graphs and Networks, the Picard–Lefschetz Theory, and Feynman Algorithms (Springer, 1975) by Solomon Lefschetz.
(i) Electric currents $J$ are 1-cycles: $\partial J=0$.
(ii) Voltages $V$ are 1-coboundaries: $V=-dU$ (U is the electrostatic potential).
(iii) There exists a duality relation between electric currents and voltages: $\langle V\vert J\rangle=0$.
(iv) If the electrical current is connected, then we get $\beta^0=1$ for the zeroth Betti number. In the general case, $\beta_0$ is equal to the number of connectivity components of the electrical circuit.
(v) If the electrical circuit has $s_0$ nodes and $s_1$ connections, then the Euler characteristic is given by $\chi=s_0-s_1$.
(vi) This yields the first Betti number $\beta_1=\beta_0-\chi$.
(vii) The space of electric currents is a linear space of dimension $\beta_1$.
(pg. 1020)

*

*Homology describes the geometry of the electric circuit; in particular, the first Betti number is equal to the number of essential loops (also called 1-cycles).

*Cohomology describes the physics of the circuit (i.e., cohomology describes voltage and hence the electric currents, by Ohm's law).

*There exists a crucial duality relation between homology and cohomology which reflects the influence of the geometry of an electrical circuit on its physics (based on the duality relation (22.17) below).
$dU(J)=U(\partial J)$ for all 1-chains $J$. (22.17)
C) Ch. 23. The Electromagnetic Field and the de Rham Cohomology:
(pg. 1027)
De Rham cohomology reformulates and generalizes the fundamental theorem of calculus due to Newton and Leibniz to differential forms on manifolds. In terms of physics, this describes the existence of potentials. The key role is played by Poincare's cohomology rule and the generalized Stokes integral theorem.
Edit (1/18/21): For a more combinatorial, graph-theoretic presentation, see the pdf "Discrete Geometric Analysis" and the slides “Discrete Geometric Analysis: Graph theory from the view of analysis and geometry” by  Toshikazu Sunada. The slides go more into the history of these ideas.
A: The Jónsson–Tarski duality between Boolean algebras with operators (in particular, modal algebras) and general frames. (A variant of this, called Esakia duality, has topological frames instead of general frames. There is also an analogous duality of Heyting algebras and intuitionistic frames, which I never remember whose name it bears.) This duality is the basis of the Kripke semantics for modal, intuitionistic, and other nonclassical logics.
A: The Curry–Howard isomorphism between typed $\lambda$-calculus and intuitionistic proofs.
A: Duality is the corner stone of the theory of Distributions
A: A very simple and important notion of duality is the following.
Start with a collection $F$ of subsets of a ground set $X$.
Now, define the blocker $F^*$ of $F$ as follows:
$$F^*=\{ X \setminus A: A \notin F \}.$$
In words, we take the complements of all sets not in $F$.
This notion is very important in combinatorial optimization and polyhedral combinatorics. It is also a simple manifestation of Alexander duality from algebraic topology.

Addendum (Adam Bjorndahl):
This construction can be viewed as a generalization of the quantifier duality
$$\forall \equiv \lnot \exists \lnot.$$
As above, fix a set $X$. For $F \subseteq 2^{X}$, define the formula $(\text{F}x) \ \phi(x)$ to mean that
$$\{x \in X : \phi(x)\} \in F.$$
So $(\text{F}x) \ \phi(x)$ might be read "for $F$-many $x$, property $\phi$ holds". Three special cases deserve some attention.

*

*When $F = \{X\}$, we recover the usual "for all" quantifier. Succinctly, $\forall = \{X\}$.


*Dualizing, we obtain
$$\lnot (\text{F}x) \lnot \phi(x) \iff \{x \in X : \lnot \phi(x)\} \notin F;$$
thus if $A = \{x \in X : \phi(x)\}$, we have
$$\lnot (\text{F}x) \lnot \phi(x) \iff A \in F^{*} \iff (\text{F}^*x) \phi(x),$$
where $F^*$ is the blocker of $F$.


*Finally, if $U \subset 2^{X}$ is an ultrafilter on $X$, then
$$\lnot (\text{U}x) \lnot \phi(x) \iff (\text{U}x)\phi(x),$$
which exhibits ultrafilters as self-dual quantifiers, a perspective I find appealing.
A: Disclaimer: the following is probably a special case of the more abstract notions in other answers above like this one. But I think it still worthwhile to expand on it a bit.
There is a notion of duality in symmetric monoidal categories. The concept is explained for example in "Duality, Trace and Transfer" by Dold and Puppe or here. All endomorphisms of dualisable objects have traces, represented as endomorphisms of the unit object. Typical examples:
In the category of vector spaces $(\mathcal{Vect},\otimes,\Bbbk)$,
a vector space is strongly dualisable if and only if it is finite-dimensional ($V$ strongly dualisable implies $(V^*)^*\cong V$, see also this answer).
For modules over a ring, strongly dualisable means finitely generated projective.
 The trace of an endomorphism is the usual trace. Note that endomorphisms of infinite-dimensional vector spaces can have a trace, but for example, the identity typically hasn't. Similar dualities exist in other tensor categories, for example, categories of representations.
In the stable homotopy category of spaces, all finite CW complexes
(finite CW spectra) have a strong dual, the Spanier-Whitehead dual. The trace of an endomorphism is its Lefschetz number, represented as an endomorphism of the sphere spectrum. Given a cohomology theory (like singular cohomology, $K$-theory or bordism),
the homology of a dualisable space is isomorphic to the cohomology of its dual.
In the case of compact manifolds, the Spanier-Whitehead dual is the Thom space of the stable normal bundle (or rather, the Thom spectrum of normal bundles), and Spanier Whitehead duality translates
into Poincaré duality (of course, there are many other proofs/explanations of Poincaré duality). Comparing the trace in the homotopy category with traces in cohomology, one gets (versions of the) Poincaré-Hopf theorem and the Lefschetz fixpoint theorem.
A: In control theory there exists the duality controllability and observability. It is very well understood in the context of linear control theory, not so much for nonlinear systems. It is related to the linear space duality between vectors and functionals, but more work in understanding it from a more general perspective would be welcome.
A: The Dual Graph of a plane graph $G$ is a graph that has a vertex corresponding to each face of G, and an edge joining two neighboring faces for each edge in $G$.
A: How about the duality between proofs and models?
A: Cartier and Pontryagin dualities.
Given a commutative nice enough topological group $G$, its Pontryagin dual is $Hom(G, \mathbb{S}^1)$. This functor provides an anti equivalence between the category of compact abelian topological groups and discrete commutative topological groups. 
By analogy, for a commutative algebraic group $G$ we can define its Cartier dual $Hom(G, \mathbb{G}_m)$. This functor provides an antiequivalence between the category of commutative affine groups and commutative group ind-finite schemes. One may further generalize to get similar statements for group stacks and abelian varieties by taking maps into $B\mathbb{G}_m$.
A: Combinatorial reciprocity is a kind of duality between the enumeration
of two related classes of combinatorial objects. The prototypical
example is for fixed $k\geq 0$ the polynomial $p_k(n)={n\choose k}$ (a
polynomial in $n$ of degree $k$). It counts the number of ways to
choose $k$ objects without repetition from $n$ objects, without regard
to order. The number $(-1)^kp_k(-n)$ counts the number of ways to
choose $k$ objects with repetition from $n$ objects, without regard
to order. This example has been vastly generalized. See for instance
the book by Beck and
Sanyal. For a "categorification" of some of this theory in terms of
commutative and homological algebra, see for instance Sections 1.8 and 1.12 of this book.
A: The (1) Fourier transform, (2) mirror symmetry, (3) electric-magnetic duality, and the (4) Pontrjagin and (5) Langlands dualities of Lie groups are all seen to be interrelated by the proposal of Strominger-Yau-Zaslow for mirror symmetry and the work of Kapustin-Witten (foreshadowed by Montonen-Olive) framing the geometric Langlands program in physical terms.
A: There are various dualities arising in elementary logic:


*

*the duality between $\forall$ and $\exists$, as
expressed by the validity $$\neg\forall x\ \neg\varphi(x)\iff
 \exists x\ \varphi(x).$$

*the duality between $\wedge$ (and) and $\vee$ (or), as
expressed via the de Morgan laws $$\neg(p\wedge q)\iff
 (\neg p)\vee(\neg q).$$

*the duality in modal logic between possibility and
necessity, as expressed via
$$\neg\Diamond\varphi\iff\square\neg\varphi,$$
(that is: $\varphi$ is not possible if and only if $\neg\varphi$ is necessary), a principle 
which has manifestations for any of the diverse
interpretations of these modal operators satisfying this
equivalence.
Each of these dualities arises in the conjugation of one
logical quantifier or operation with $\neg$.
A: Projective geometry.  Is that the first use of the term "dual" in mathematics?
A: Serre duality
Grothendieck duality
Verdier duality
A: The duality between measure and category in the set theory of the reals.
A: Umbral compositional inversion is a type of duality related to multiplicative and compositional inversion of functions and to matrix inversion, which is very useful in deriving algebraic relations and other identities among polynomial sequences important in number theory, special functions, and enumerative combinatorics as well as operator calculi.
If a pair $p_n(x)$ and $q_n(x)$ of polynomial Sheffer sequences is an umbral inverse pair (UIP), then
$$p_n(q.(x))= x^n=q_n(p.(x)),$$
where, e.g., $p_n(q.(x))=\sum^n_{k=0} \; p_{n,k} \cdot q_k(x).$
This implies that the pair of lower triangular matrices comprised of the coefficients of these polynomial sequences are an inverse pair.
An important UIP of binomial Sheffer sequences are the Bell / Touchard polynomials, comprised of the Stirling numbers of the second kind, and the falling factorials, comprised of the Stirling numbers of the first kind. An important UIP of Appell Sheffer sequences are the Bernoulli polynomials and the reciprocal integer polynomials.
For a binomial Sheffer sequence of polynomials, defined by the exponential generating function
$$e^{h(t)x}=e^{p.(x)t},$$
with $h(0)=0$ and $(p.(x)^n)=p_n(x)$, the umbral compositional inverse Sheffer sequence is given by
$$e^{h^{(-1)}(t)x}=e^{q.(x)t},$$
where $h$ and $h^{(-1)}$ are a compositional inverse pair. E.g., $(h(t),h^{-1}(t))=(e^t-1,\ln(1+t))$ generate the Bell and falling factorial pair.
For an Appell Sheffer sequence, defined by the e.g.f.
$$f(t)e^{x \cdot t}= e^{u.(x)t},$$
with $f(0)=1$, its umbral compositional inverse Appell sequence is given by
$$\frac{1}{f(t)}e^{x \cdot t} = e^{v.(x)t}.$$
For example, $f(t)=\frac{t}{e^t-1}$ for the Bernoulli polynomials $B_n(x)$, so
$$e^{v.(x)t}=\frac{e^t-1}{t}e^{xt}$$ for the UI dual, the reciprocal integer polynomials,
$$v_n(x)=\frac{(1+x)^{n+1}-x^{n+1}}{n+1}.$$
Consequently, using the simple convolution properties of Appell sequences and the UI relation,
$$ \frac{(1+B.(x))^{n+1}-B.(x)^{n+1}}{n+1} = \frac{(B.(1+x))^{n+1}-B.(x)^{n+1}}{n+1} = x^n = \frac{d}{dx} \frac{x^{n+1}}{n+1},$$
so we can formally identify for any power series
$$S(B.(1+x))-S(B.(x)) = S^{'}(x).$$
There are many interesting and useful dual operators associated with these UIPs (e.g., the first derivative and the forward finite difference operator and its inverse are related to the Bernoulli UIP), and the umbral inverse property alone is often very useful in giving simple, concise derivations of properties of the polynomials and their coefficients. Interweaving the two types of inversions, multiplicative and compositional, a simple formula for the Bernoulli polynomials and their associated base numbers, the Bernoulli numbers, in terms of the Stirling numbers is easily derived in my blog post Compositional Inverse Operators and Sheffer Sequences.
A: I'm surprised no one has mentioned this one yet. Duality between simply connected Riemannian symmetric spaces of compact and noncompact type. I personally think this is a perfect example because it demonstrates how useful duality can be in mathematics:

*

*First of all, this is a genuine duality, meaning, it gives a bijective correspondence between isometry classes of simply connected symmetric spaces of compact type on one side and symmetric spaces of noncompact type on the other (these ones are automatically simply connected). And, surely, $M^{**} \cong M$.

*$M$ and $M^*$ share several basic properties in common: dimension, rank, (the identity component of the) isotropy subgroup of the isometry group, which in this situation is the same as the holonomy group. There is also a bijective correspondence between totally geodesic submanifolds of $M$ and $M^*$. Duality respects de Rham decomposition and thus irreducibility.

*In particular, if you want to classify all (simply connected) symmetric spaces, it suffices to classify irreducible ones of just one of the two types, since you get all irreducible spaces of the other type for free by duality, and any (simply connected) symmetric space will then be the product of irreducible ones of these two types and a Euclidean space. If I'm not mistaken, this is exactly what Élie Cartan did in his classification of symmetric spaces in 1926: he obtained a list of compact irreducible ones, and the rest followed.

*Finally, since symmetric spaces of (non)compact type are quotients of (non)compact semisimple Lie groups, the methods one uses to study spaces of these two types are quite different. You may have a geometrical problem (e.g., classify polar or cohomogeneity-one isometric actions) that is easier to solve on one of the two types, and then duality enables you to transfer some of the results to spaces of the other type to obtain at least a partial solution there.

A: Galois connections (nLab, Wikipedia). This is really just an adjunction between one category and the opposite of another, where these categories are preorders. A Galois correspondence is when this adjunction is an equivalence of categories.
Stone duality (nLab, Wikipedia). This is best explained by the linked page, but one I will point out is that one has as a small part of this duality, $FinSet \simeq FinBool^{op}$ (the category of finite sets is equivalent to the opposite of the category of finite boolean algebras), which has as a corollary, the category of Stone spaces is equivalent to that of profinite sets.
There is the nLab page duality, but one can see by searching the nLab, there are a number of other pages that people might find useful.
A: What would be useful here is a list of mechanisms lying behind these appearances of duality. So we have (at least)

*

*Duality pairing

*Dualizing object

*Maximal fixed subcategories of an adjunction

*Arrow reversal

Then we could look at any relations between these mechanisms, such as between 2 and 3, maps into a dualizing object form the functors for an adjunction.
Atiyah in his talk Duality in Mathematics and Physics says

"Fundamentally, duality gives two different points of view of looking at
the same object. There are many things that have two different points
of view and in principle they are all dualities."

So perhaps we need
5 . Something is seen in two different ways
The Dynkin diagram for $SL_n$ is a string of $n-1$ dots, we can view it from either end as point, line, plane, etc. Put another way, the symmetry of the diagram corresponds to an outer automorphism which account for the duality of projective geometry.
I wonder if 'deeper' dualities come from more intricate processes of seeing something from two points of view. Frenkel gives a very accessible talk What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common? where he explains that the duality of Geometric Langlands arises from compactifying a 6d quantum field theory in two different ways onto 2d surfaces.
A: I think that the obvious one between spaces (topological, differentiable, algebraic, etc.) and the rings of structure preserving functions on them should be mentioned. 
A: In the study of convexity and convex polyhedra there are three (related) important notions of duality

*

*Polar duality
This is a map assigning to every convex set $K$ containing the origin its polar dual: $K^*$ which is the set of all points whose inner product with every point in $K$ is at most 1.
On polytopes it induces an order reversing map on the face lattices. This operation has subtle relations to mirror-symmetry and Koszul duality.
(Web sources: 1; 2; 3; 4; 5)


*Gale transform
This is an operation to move from $n$ points in $\mathbb R^d$ to $n$ points in $\mathbb R^k$ where $k=n-d-1.$ It is especially useful if the original $n$ points are in convex position to study the convex polytope they define.
(Web sources: 1; 2; 3; 4)


*Linear programming duality
This is an operation to move from a linear programming problem to a dual problem which have the same solution.
(Web sources: 1; 2; 3)
A: The duality between projective modules and injective modules, also the duality between divisible abelian groups and free abelian groups.
A: The color-kinematics duality underlying connections among various quantum field theories and governed by relations between Feynman diagrams for scattering amplitudes and the associahedra is a hot topic these days:
See the presentation "Scattering Amplitudes and the Associahedron" by Bai,
"The Duality Between Color and Kinematics and its Applications" by Bern,
and
"Combinatorics and topology of Kawai-Lewellen-Tye relation" by Mizera.
(In Mizera, a relation between two sets of scattering matrices and the KLT relations is approached through compositional inversion that gives the aerated Catalan numbers that are related to tree level Feymman diagrams and, of course, are the number of vertices of the associahedra, whose combinatorics govern the compositional inversion of power series and, therefore, have connections to the Legendre transform, which also can be couched as a relation between compositional inverses, and to solutions of the inviscid Hopf-Burgers' equation, which has a connection to conservation laws of integrable hierarchies. This connection to the Catalan numbers and combinatorics of the associahedra is perhaps foreshadowed in a paper by Bessis et al. "Quantum field theory techniques in graphical enumeration", in which the authors unknowingly wrote down an o.g.f. for the Catalan numbers, OEIS A000108, and one by Kreimer and Yeats "Diffeomorphisms of quantum fields" in which the authors unknowingly listed the partition polynomials for compositional inversion for power series, OEIS A133437.)
(Related MO-Qs: Q1. Guises of the Associahedra and Q2. An Intriguing Tapestry)
A: Differential geometry: Eigenvalues of Laplace operators  $\Leftrightarrow$ length of closed geodesics
representation theory: irreducible representations $\Leftrightarrow$ conjugacy classes in a group
Number theory: primes $\Leftrightarrow$ zeros of $L$ functions
Quantum mechanics: particles $\Leftrightarrow$ waves
Argument principle in complex analysis: contour integrals $\Leftrightarrow$ residues
Index theory: topological index $\Leftrightarrow$ analytic index
Algebraic geometry:  algebraic cycles $\Leftrightarrow$ motives
Most of them can be found in: www.claymath.org/cw/arthur/pdf/52.pdf
Trace formulas like Poisson summation formula, Arthur's trace formula, Selberg's trace formula, Gutzwiller trace formula, Lefschetz trace formula, Weil's explicit formula quantify these relations.
There is always a sort of Fourier uncertainty involved, so a one-to-one correspondence between "geometric objects" and "spectral objects" is not available, except perhaps for the symmetric group $S_n$ via Young diagrams.
A: I am personally fond of matroid duality.  Let $M$ be a matroid with ground set $E$.  The dual of $M$ is the matroid with ground set $E$ and whose bases are the complements of bases of $M$.  It is easy to verify that the dual of $M$ is indeed a matroid and we immediately have that $M^{dd}=M$.
Matroid duality illustrates that deletion and contraction are actually dual operations.  That is, deletion corresponds to contraction in the dual and vice versa.
It also nicely generalizes duality for planar graphs.  That is, if $G$ is a planar graph, and $M(G)$ is the cycle matroid of $G$, then $M^d(G)=M(G^d)$ (here, $G^d$ is the planar dual of $G$).
Finally, here is a proof for free of Euler's Formula via matroid duality.  Let $G$ be a connected planar graph with edge set $E$.  Suppose that $G$ has $v$ vertices, $e$ edges and $f$ faces.  Let $r$ be the rank function of the cycle matroid of $G$ and let $r_d$ be the rank function of the dual of the cycle matroid of $G$.  Then
\[
e=r(E)+r_d(E)=(v-1)+(f-1).
\]
A: Koszul duality is a useful duality. For example, one can cite


*

*Koszul duality of quadratic algebras (due to Priddy) which is related to inversion of formal power series.

*Koszul duality of quadratic operads (due to Ginzburg and Kapranov) which is related to reversion of formal power series or plethystic reversion.

*Koszul duality of cyclic quadratic operads (due to Getzler and Kapranov) which is related to Legendre duality and Legendre transform.
One can see (1 and 2) that Koszul duality is often related to the notion of inversion $g \mapsto g^{-1}$ in a group.
A: This answer has a heavy bias towards logical structures. The simplest notion I know is order-theoretic duality.

*

*The dual of an order is the inverse relation of the order (less-than vs. greater-than, subset vs. superset)

*Greatest lower bounds and least upper bounds (minimum vs. maximum, intersection vs. union, conjunction vs. disjunction)

*Bottom and top

*Least and greatest fixed points

*Additive and multiplicative maps

In structures containing negation, we have De Morgan duality, such as the examples from logic given by Joel David Hamkins.
I do not know if 'duality' is the right term, but I think of adjunctions as duals too. To add to the answer of David Roberts:

*

*Conjunction and implication (both with one argument fixed) are adjoints

*Existential and universal quantification are adjoints to a certain form of substitution

*Strongest postconditions and weakest liberal preconditions in programming language semantics

*Sets of models and sets of formulae

*A lattice and its image under a closure operator

In settings with a notion of time, there are temporal dualities from the interaction of the past and the future. There are several examples in temporal and modal logics.
Some representation theorems for lattices are ancestors of dualities. For example, Stone's representation theorem for Boolean algebras is now usually referred to as a duality. There are various dualities relating families of lattices with families of discrete structures.

*

*Complete, atomic, Boolean algebras and powersets [Lindenbaum and Tarski]

*Finite distributive lattices and finite posets [Birkhoff]

*Completely distributive, algebraic lattices and posets [Raney, others I cannot recall]

*Boolean algebras with operators and sets with relations [Jónsson and Tarski]

*Distributive algebras with operators and ordered sets with relations [Gehrke and Jónsson (though there may be earlier work)]

The list goes on. Such results are sometimes called discrete dualities. There is much recent work on discrete duality in terms of what are called canonical extensions. These duality results often include a topological component.

*

*Boolean algebras and Stone spaces [Stone]

*Distributive lattices and Priestley spaces [Priestley]

*Heyting algebras and Esakia spaces [Esakia]

*Topological representations of arbitrary lattices [Urquhart]

*Extensions of Stone and Priestley duality to lattices with operators

*Dualities arising in Modal logic [Goldblatt]

One 'analogy between analogies' is that of a dualising object. The term schizophrenic object has also been used in this context.
Porst and Tholen's article Concrete Dualities discusses some of these and other dualities and the connection to adjunctions. Other references are Peter Johnstone's book Stone Spaces and Clarke and Davey's book Natural Dualities for the Working Algebraist.
A: Finite-dimensional linear spaces. A particular feature in this case is that the (algebraic) dual of a finite-dimensional vector space, namely the space of linear maps from the vector space into the base field, is isomorphic to the original space (since it is of the same dimensionality) but not canonically so. In contrast, the bi-dual (the dual of the dual) is canonically isomorphic to the original space, and so may be identified with it.  
A: A simple kind of duality in logic is between implication and set-theoretic inclusion, which explains why the horseshoe $\supset$ is found in both contexts. The most natural way to think about it, is the reverse of the actual  usage:
A $\supset$ (implies) B if A $\subseteq$ (is a subset of) B
So for instance, 
x is a person implies x is a mammal, since the set of people is a subset of the set of mammals.
A: The contravariant powerset functor $P : C^{\text{op}} \rightarrow C$ is the canonical example of duality.[1]
There are several intereresting duality principles which amount to application of the contravariant powerset functor. The duality between products and coproducts in category theory can be seen as application of the contravariant powerset functor on the definition of the coproduct. $P(A+B) \cong PA \times PB$ and
\begin{gather*}
P([f,g] : A+B \rightarrow D) = \\
\langle Pf,Pg\rangle : PD \rightarrow P(A + B).
\end{gather*}
Similarly for injections to the coproduct:
\begin{gather*}
P(i_1 : A \rightarrow A+B) = \\
\pi_1 : P(A+B) \rightarrow PA.
\end{gather*}
By expanding $P(A+B) \cong PA \times PB$ and using the coproduct laws, these are seen to be the product laws, which justifies use of that notation for products above, where you often suppress explicit notation for the isomorphism. The other projection is similar. This kind of application of the contravariant powerset functor can be done for all finite colimits.
For exponentials, the situation is much different. Trying to find operation $B \setminus A$ (not the set subtraction, but close) such that the contravariant powerset functor produces an exponential, something along the lines of $P(B \setminus A) \cong PA \Rightarrow PB$ is a cause of much confusion about duality, since coexponentials are not very natural concept [2] and can cause havoc when combined with, say, a topos. Inverting the arrows in $C^{op}$, in essence $B \setminus A \cong B \Rightarrow A$, seems obvious solution, but there aren't necessarily exponentials in $C^{op}$, and (speculation:) the notion of exponential as internalization of hom-sets somehow inverts as well. There are several important notions of topology (open sets, bounded sets) that seem suitable for such definition in terms of exponential $PA \Rightarrow PB$. This also interacts with problems defining a left adjoint to a partially applied coproduct functor $F \dashv A + -$ (which might also be called subtraction if it exists).
I can see two possible outcomes for such construct. Either the power set hierarchy expands without limit and iterating the exponential (which is similar to iterating the power set) produces ever larger categories. Or the possibilities for such expansion are limited, and some principle, e.g. Rice's theorem, limits the expressive power of repeated towers of exponentials.
[1] Lawvere, Rosebrugh: "Sets for mathematics"
[2] Crolard: Subtractive logic
A: Polytope duality
This relates both, the duality of convex sets (polar dual) and duality of lattices (via the face lattice of a polytope).
In the case of 3-dimensional polyhedra, it is related to the duality of planar graphs.
A: The transposition of Young tableaux yields a kind of duality, internal to the set of partitions of an integer $n$.
A: From "Hilbert series and Gelfand duality" by Vadim Schechtman (CiteSeerX, arXiv:0908.4533):
In their great work on spherical functions ... Berezin and Gelfand wrote:
”... there exists a deep duality between the function ... giving the law of multiplication in the center of the [infinitesimal] group ring [of a semisimple Lee group]and the function ... giving multiplication of representations.
... an analogous duality exists between matrix elements of ... an irreducible
representation of the group SU(2) ... and the so-called ”Clebsch-Gordan coefficients”...
Another example of such a duality are the formulas of Gelfand-Tsetlin for
matrix elements of irreducible representations of the algebra of complex matrices
with trace 0 and the formulas for coordinates in the group of unitary matrices...
In all of these cases the duality consists in the fact that functions of discrete
arguments satisfy finite difference equations analogous to differential equations
satisfied by functions of real variables that correspond to them.”
The second of the above examples may be expressed by saying that we have
a duality between the classical orthogonal polynomials (Jacobi etc.) and their
discrete analogues (Hahn etc.). (In fact, all of the above examples admit a similar
reformulation.)
The main purpose of the present note is to propose an example illustrating that
exactly this type of dual polynomials appears in certain Hilbert series.
