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.
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".
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 $\mathbb{C}$.