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 $\mathbb{C}$.