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 [Jonnson Jónsson and Tarski]
- Distributive algebras with operators and ordered sets with relations [Gehrke and Jonnson 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.