Yes, a finite ring $R$ is a finite direct sum of local finite rings.  As a first step, for each prime $p$ there is a subring $R_p$ of $R$ corresponding to the elements annihilated by the powers of $p$; $R_p$ is then an algebra over $\mathbb{Z}/p$.  This step generalizes to maximal ideals:  For each maximal ideal $m$, $R_m$ is the subring of elements annihilated by $m^n$ for some $n$, and $R$ is the direct sum of these subrings, which are local rings.

It is not difficult to write down a rough partial classification of of local finite rings.  If $R$ is local with maximal ideal $m$, it is an algebra over the finite field $F = R/m$.  If you choose a basis $x_1,\ldots,x_n$ for $m/m^2$, then $R$ is a quotient of the polynomial ring $F[\vec{x}]$ in which only finitely many monomials are non-zero.  You can make a diagram of these non-zero monomials; they can be any order ideal in the $n$-dimensional orthant.  Or, in basis-independent form, $R$ has a length, which is the largest nonvanishing power of $m$, and each $m^k/m^{k+1}$ is some quotient of the $k$th symmetric power of the generating vector space $V = m/m^2$.

However, after that there may be linear dependencies among the monomials.  Informally, there will be an endless stream of partial results and there will never be a complete classification when the length of the local ring is 3 or more.  To see this, suppose that $m^4 = 0$, and suppose that $m^3$ is only one dimension shy of $S^3(V)$.  Then the ring is defined by an arbitrary symmetric trilinear form in $V$.  These make a "wild" sequence of algebraic varieties, in the same sense that people say that the representation theories of certain rings are wild.  For instance, I think (not quite sure) that it is NP-hard to determine when two such trilinear forms are equivalent.  NP-hardness is not by itself rigorously equivalent to no classification, but informally the classification is an intractable mess.

If the nonvanishing monomials in $R$ are linearly independent, then it is a toric local ring.  Toric local rings are certainly a tractable class of finite rings.