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 $P^*$,
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 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".