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. 



Examples in mirror symmetry that can be understood.
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.


