From one point of view, the classification of finite non-commutative rings or even finite commutative rings is wild and tons of things can happen. From another point of view, finite rings are highly restricted and not all that much can happen.

By the Chinese remainder theorem, every finite ring is unique direct sum of $p$-primary finite rings, i.e., rings whose cardinality is a power of some prime $p$. Such a ring is an algebra over $\mathbb{Z}/p^k$ for some $k$, and the finiteness condition is then equivalent to the statement that this algebra is Artinian and finitely generated. We can look at the case $k=1$ for a start, because then you get a finite-dimensional algebra over a field. Rings over $\mathbb{Z}/p^k$ for higher $k$ will look similar to these algebras.

A finite-dimensional algebra has Jacobson radical and a maximal semisimple quotient if you quotient by that radical. By the theorems of Wedderburn and Artin-Wedderburn, the semisimple quotient is a direct sum of matrix algebras over a finite field. So that wraps up the semisimple examples.

Let's suppose at the other end that the semisimple quotient is a finite field $\mathbb{F}_q$. Any such algebra is a finite-dimensional quotient of free polynomial algebra $\mathbb{F}_q\langle x, y, \ldots \rangle$. In fact it will be a quotient of the truncation defined by killing all monomials of degree $k$, for some $k$. Such a truncation is already an interesting example, e.g. the algebra spanned by $1,x,y,x^2,y^2,xy,yx$, with all larger monomials set to 0. The wild part of the classification is that a general quotient of this type can be very complicated, because it is defined by an ideal or a truncation in a very complicated position. (It is like cutting a riser pipe: The cut can be smooth or very jagged.)

The general finite-dimensional algebra will be a combination of these various ideas. That is kind-of glib because you can get things like truncated path algebras, but it is true.