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.
