Examples in mirror symmetry that can be understood. It seems to me, that a typical science often has simple and important examples whose formulation can be understood (or at least there are some outcomes that can be understood).  So if we consider mirror symmetry as science, what are some examples there, that can be understood? 
I would like to explain a bit this question. If we consider the article 
"Meet homological mirror symmetry" http://arxiv.org/abs/0801.2014
it turns out, that in order to understand something we need to know huge amount of material, including $A_{\infty}$ algebras, Floer cohomology, ect.
Here, on the contrary, is an example, that "can be understood" (for my taste): According to Arnold, the first instance of symplectic geometry was "last geometric theorem of Poincare". 
This is the following statement:
Let $F: C\to C$ be any area-preserving self map of a cylinder $A$ to itself, that rotates the boundaries of $A$ in opposite directions. Then the map has at least two fixed points.
(this was proven by Birkhoff http://en.wikipedia.org/wiki/George_David_Birkhoff)
So, I would like to ask if there are some phenomena related to mirror symmetry that can be formulated in simple words.
Added. I would like to thank everyone for the given answers! I decided to give a bit of bounty for this question, to encourage people share phenomena related to mirror symmetry that can be simply formulated (or at least look exciting). Since there are lot of people in this area I am sure there must be more examples. 
 A: A toy model for mirror symmetry is the following. Consider a real manifold (not necessarily compact) $B$ with an atlas of affine coordinates, i.e. such that the change of coordinate maps are of the type $x \mapsto Mx  + b$ where $M$ and $b$ are constant. Then the tangent bundle $TB$ has natural complex coordinates given by $z = x +i y$, where $y$ are coordinates on the fibre. If further one assumes that $\det M = 1$ then $TM$ also has a nowhere vanishing holomorphic $n$-form. On the other hand $T^*B$ has it's usual symplectic structure. So $TB$ and $T^*B$ can be thought of as being mirror. One can also twist the complex structure on $TB$ with a B-field. One can go a bit further too, in fact suppose $\gamma$ is an affine curve in $B$ (i.e. a straight line in affine coordinates). Then one can lift $\gamma$ to a Lagrangian submanifold $L_{\gamma}$ of $T^*B$ by adding the annihilator of $\gamma'(t)$ in the fibre at $\gamma(t)$. On the other hand the same curve lifts to a complex object in $TB$ by adding the line generated by $\gamma'$ inside the fibre $T_{\gamma} B$. If the affine structure is also integral (i.e. $M$ and $b$ have integral coefficients), then one can also partially compactify by taking a lattice $\Lambda \subset TB$ and its dual $\Lambda'$ and then form torus bundles $X = TB / \Lambda$ and $X^* = T^*B / \Lambda'$. This picture is too simple to work in the compact case, but it is expected that actual mirror symmetry is a perturbation of this. What I just described is the SYZ approach to mirror symmetry. 

I like this example. Consider a surface $V$ in $(\mathbb{C}^{\ast})^2$ given by some Laurent polynomial $p(z)$, then the hypersurface $X$ defined by $xy = p(z)$ is Calabi-Yau. Its mirror $\check X$ can be constructed by taking the Newton polygon $\Delta$ of $p$ and then consider the toric variety defined by the cone over $\{1\}\times\Delta$ in $\mathbb{R}^3$. $\check X$ is a resolution of this toric variety obtained from some subdivision of $\Delta$. Now, the surface $V$ (the one we started from) has a "tropical amoeba". This can be thought of a graph in $\mathbb{R}^2$ which is the limit (in some sense) of the image of $V$ under the standard torus fibration $(\mathbb{C}^{\ast})^2 \rightarrow \mathbb{R}^2$. The interesting thing is that this graph gives a subdivision of $\mathbb{R}^2$ which is dual to the subdivision of $\Delta$ (this is related Gil Kalai's answer). More over this graph is also the locus of singular fibres of a Lagrangian torus fibration defined on $X$. Such a Lagrangian fibration induces on the base an affine structure as I said previously. A construction such as the one I described above can be used to construct many Lagrangian $S^3$'s in $X$ over the bounded regions defined by the graph. The mirror of these objects are the divisors $\check X$ corresponding to interior integral points of $\Delta$, or better, line bundles supported on such divisors. There are some interesting correspondences between intersection points of these spheres and cohomology of the line bundles, even without getting into $\mathcal{A}^{\infty}$ constructions. 
A: I'm not sure if this is what you're looking for, but the paper "Mirror symmetry and Elliptic curves" by R. Dijkgraaf might be provide a good example.
The example in that paper concerns the mirror of an elliptic curve. They have two moduli parameters, their complex moduli and their Kahler moduli. Mirror symmetry in this case simply states that the mirror of some $E_{\tau, \omega}$ is $E_{\omega,\tau}$ i.e. you simply switch the two moduli parameters.
A: Here is my biased view of a simple example:  the two-torus.
Everything I know about homological mirror symmetry
stems from this example.
Because the example is one-dimensional, a symplectic form
is just an area form, and Lagrangians are simply
curves, and the holomorphic maps which are part of the Fukaya
category are simply topological disks.  (By uniformization of
Riemann surfaces, there is one holomorphic map for each topological
disk satisfying the appropriate boundary conditions.)
Even better, you can go to the universal cover, which is $R^2,$
and just draw Lagrangians as straight lines with rational
slope.  The holomorphic disks which determine
compositions in the category are simply triangles. 
On the mirror side, we're talking about a complex two-torus, or
elliptic curve.  A typical object would be a line bundle on
the elliptic curve, such as the theta line bundle, whose
sections are theta functions, once we lift them up to the
complex plane.
The two-torus is circle-fibered over a base circle,
and the elliptic curve is circle-fibered by the dual circle
(i.e., $U(1)$ local systems on the original circle).
This is called T-duality, and it explains how to construct the
mirror equivalence going from Lagrangians to line
bundles, or vice versa.  For example, the Lagrangian $\{y=0\}$
represents a family of trivial $U(1)$ local systems, corresponding to
the trivial holomorphic line bundle whose sections are just holomorphic
functions.  The Lagrangian $\{y=nx\}$ corresponds to a line bundle
of degree $n$.  After making these definitions, one checks
that compositions match up.
A: You need the machinery of triangulated categories and homological algebra to understand the mirror symmetry as it stand today, Homological mirror symmetry. But one can get an idea of mirror symmetry without delving into these concepts,I am talking about the classical picture of mirror symmetry as noticed by Physicists. i.e. Mirror symmetry as an isomorphism between the complex and Kahler moduli spaces of Calabi-Yau 3-folds.(Beware : this was first definition of mirror symmetry and even here I am overlooking the subtleties involving the large complex structure limit.) e.g. Elliptic curve (Dijgraaf), Quintic (Greene, Plesser and Candelas et al).This may give an idea of Mirror symmetry but to understand this picture properly one need an understanding of the geometry of Calabi-Yau manifolds, Variations of mixed Hodge structures, quantum cohomology and GW invariants.
Also there is a modern picture of mirror symmetry called SYZ conjecture which is more geometric and doesn't involve homological algebra and triangulated categories. But again you need the knowledge of geometry of special Lagrangian submanifolds of CY manifolds. 
A: Since you are asking for examples, you might want to take a look at the lecture notes of Mark Gross published in Calabi-Yau Manifolds and Related Geometries. In the chapter "Mirror Symmetry in Practice" the case of the quintic is worked out in some detail. 
Although one does not need to know what an $A_\infty$ algebra is, one has instead to be familiar with variations of Hodge structure and some symplectic geometry. However, as far as I know, homological mirror symmetry is actually weaker than what is believed to be true, so it does probably not hurt to see, what one can show in specific examples. You might also want to look at this book, it is written for an audience of physicists and mathematicians, but probably does not represent the most recent view on mirror symmetry.
A: My master's thesis, An Introduction to Homological Mirror Symmetry and the Case of Elliptic Curves, might provide a piece of what you're looking for. It mostly concerns itself with the symplectic side of HMS (cause I have only a very superficial knowledge of algebraic geometry), but it includes a good amount background and some history. I tried to make it a useful document for other beginners to read... though then I let it rot on JSTOR for two years before posting it on arXiv :)
Oh well, it's up there now. I hope it helps!
A: Mirror symmetry gives some remarkable connections between certain varieties. The first step in this connection is that certain homology groups have the same rank. An explicit case for mirror symmetry duals is the case coming from toric varieties. In this case, the dual objects comes from duality of polytopes. So duality of polytopes: associating the octahedron to the cube and the icosahederon to the dodecahederon is related to Mirror symmetry. 
Perhaps the very first facts about polytopes which demonstrates unexpected equalities for certain homologies can be described as follows: 
For 2-dimensional polytopes this is the following numerical fact: A polygon has the same number of edges as the dual. (Well, this is not so unexpected.)
For 4 dimensional polytope P it is the following numerical fact. Start with a 4-polytope with n vertices and e edges. Triangulate every 2-face by non crossing diagonals. Let $e^+$ be the number of edges including the added diagonals. Consider the quantity
$$\gamma (P) = e^+ - 4n . $$
It is true that for every dual pair of 4-polytopes $P$ and $P^*$, 
$$\gamma (P^*)=\gamma(P).$$ 
This is more surprising. 
For example, let P be the 4-dimensional cross polytope and Q be the 4-dimensional cube. P has 8 bertices 24 edges and all the 2-faces are triangles so $\gamma (P)=24~-~4\cdot 8~=~-8$. The 4-cube Q has 16 vertices, and 32 edges and it has also 24 2-faces which are squares, so $e^+(Q)=56$. $\gamma (Q)=56-64 = -8$. Voila!   
This reflects some properties of toric varieties (unexpected equalities between Hodge numbers) which express (sort of the 0-th step of) mirror symmetry.
Related blog post: a mysterious duality relation for 4-dimensional polytopes; Related papers: V. Batyrev and L. Borisov, Mirror duality and string-theoretic Hodge numbers; V. Batyrev and B. Nill, Combinatorial aspect of mirror symmetry. Here is a lecture by B. Nill.

Another manifestation of mirror symmetry of combinatorial nature, that can be formulated in simple words, is in terms of typical shape of various classes of partitions. I mentioned it in a remark above and let me quote a description taken from my adventure book.

A partition is just a way to write a
  number as a sum of other numbers. Like
  9=4+2+1+1+1. Partitions have attracted
  mathematicians for centuries. Among
  others, the famous Indian
  mathematician Ramanujan was well known
  for his identities regarding
  partitions. And now enters another
  idea, baring the names of Ulam,
  Vershik, Kerov, Shepp and others who
  studied partitions as stochastic
  objects. In particular, it was
  discovered that "most" partitions, say
  of a number n, come in a "typical
  shape".
The emergent picture drawn by Okounkov
  and his coauthors goes very roughly
  like this: an "algebraic variety" (a
  manifold of some sort) that takes part
  in a certain string theory is related
  to a class of partitions, and when we
  consider the typical shape of a
  partition in the class this gives us
  another algebraic variety, and - lo
  and behold - the typical shape IS the
  mirror image of the original one. The
  mirror relations translate to
  asymptotic results on the number of
  partitions, somewhat in the spirit of
  the famous asymptotic formulas of the
  mathematicians Hardy and Ramanujan for
  p(n)- the total number of partitions
  for the number n.

As mentioned in the comments I am not sure about good references to this connection between mirror symmetry and limit shapes of classes of partitions. The 2003 paper  Quantum Calabi-Yau and Classical Crystals by Andrei Okounkov, Nikolai Reshetikhin, and Cumrun Vafa describes this connection in Section 2.3 called "mirror symmetry and the limit shape".
A: Here is the simplest example that I can think of...
The ordinary cohomology ring of $\mathbb{CP}^n$ is given by $\mathbb{C}[a]/(a^{n+1})$. The structure of this ring can be thought of as describing the intersection theory of subvarieties / submanifolds / linear subspaces of $\mathbb{CP}^n$. For example, the relation $a^3 \cdot a^3 = 0$ in the cohomology ring of $\mathbb{CP}^5$ reflects the fact that the intersection of two generic dimension 2 subspaces of $\mathbb{CP}^5$ is empty.
Now the quantum cohomology ring of $\mathbb{CP}^n$ is $\mathbb{C}[a]/(a^{n+1} - q)$, where we can think of $q$ as being a nonzero constant, or a formal parameter if you like. The quantum cohomology ring is a deformation (in a suitable sense) of the ordinary cohomology ring. The structure of the deformed ring now encodes "enumerative geometry" information. For example, it is a fact that given generic linear subspaces $A,B,C$ of $\mathbb{CP}^n$ of total dimension $n-1$, there is a unique degree 1 map $\mathbb{CP}^1 \to \mathbb{CP}^n$ sending the points $0,1,\infty$ to $A,B,C$ respectively. Writing $q$ as $1 \cdot q^1$, the coefficient $1$ corresponds to the uniqueness of the map, and the exponent $1$ corresponds to the degree of the map. I like to think of this as a generalization of the fact that there is a unique line passing through any two distinct points in the plane, which has been known since at least Euclid... :-)
But so far I haven't said anything about "mirror symmetry"...
Mirror symmetry says that the story I've described above is echoed by certain properties of the function $W = x_1 + \cdots + x_n + \frac{q}{x_1\cdots x_n}$ on $(\mathbb{C}^\ast)^n$. For example, the Jacobian ring of $W$, which is by definition the ring $\mathbb{C}[x_i^{\pm 1}]/(\partial_i W)$, is isomorphic to $\mathbb{C}[a]/(a^{n+1} - q)$.
EDIT: The relation between $\mathbb{CP}^n$ and $W$ goes much deeper. For another elementary(-ish) mirror symmetry statement, there is Seidel(I think?)'s proof that the derived category of $\mathbb{CP}^n$ is equivalent to the Fukaya-Seidel category of $W$. In this case these categories can be described fairly easily, without too much fancy language, via the "Beilinson quiver", which on the derived category side corresponds to the line bundles $\mathcal{O}, \mathcal{O}(1), \cdots , \mathcal{O}(n)$ and the fact that there is an $(n+1)$-dimensional set of morphisms from $\mathcal{O}(i)$ to $\mathcal{O}(i+1)$. For example, consider the morphisms from $\mathcal{O}$ to $\mathcal{O}(1)$; these are just the sections of $\mathcal{O}(1)$, which are the homogeneous degree 1 polynomials in $n+1$ variables.
On the other side, one can see the Beilinson quiver via the "vanishing cycles" $L_0, L_1, \dots, L_n$ of $W$, and the $n+1$-many morphisms above correspond to the $n+1$ intersection points between $L_i$ and $L_{i+1}$. For more on this, see the notes from Bohan Fang's talk here and this paper of Seidel.
This kind of correspondence between vector bundles and cycles, and between morphisms of vector bundles and intersections points of cycles, is a first approximation of homological mirror symmetry, or "categorical" mirror symmetry. For a better approximation, the statement is that compositions of morphisms of vector bundles correspond to "compositions" of intersection points, where these "compositions" are defined via $J$-holomorphic discs. But for the elliptic curve / symplectic torus, things are still pretty simple, and one can avoid saying the word "$J$-holomorphic disc" if one wishes. In this situation, the correspondence between compositions reduces to a correspondence between some classical facts about theta functions on elliptic curves and some very elementary observations about lines and triangles on a torus.
And finally, here is the most trivial example of mirror symmetry. Let $X$ be a point $\operatorname{Spec} \mathbb{C}$. Then the mirror of $X$, call it $Y$, is also a point. Notice that the point is a Lagrangian submanifold of $Y$. Notice that the point intersect the point is the point. On the other hand, take $\mathbb{C}$ as a $\mathbb{C}$-module. Then there is a 1-dimensional set of $\mathbb{C}$-module morphisms from $\mathbb{C}$ to $\mathbb{C}$.
A: I won't claim this means ''understanding mirror symmetry'', but if you are familiar with the derived category of coherent sheaves, then there is a consequence of Kontsevich's homological mirror symmetry that can be understood, without knowing anything about the Fukaya category:
For every symplectic diffeomorphism of the mirror $\hat X$, there is a autoequivalence of $\mathrm D^b(X)$.
Examples:


*

*If $X$ is a an elliptic curve, then $\mathrm{SL}_2(\mathbb Z)$ acts as a group of symplectic diffeomorphisms on the mirror $\hat X = \mathbb{R}^2/\mathbb{Z}^2$. There is a corresponding action of (a central extension of) $\mathrm{SL}_2(\mathbb Z)$ generated by Fourier-Mukai transform induced by the Poincare line bundle on $X \times X$, and by tensoring by a line bundle of degree one (and shifts).

*A Dehn twist corresponds to the ''spherical twist'' $\mathrm{ST}_E$ at an spherical object $E$ (see arXiv:math.AG/0001043).

*More examples have been studied by Horja, see arXiv:0103.5231.
