Is there a classification of finite commutative rings available? If not, what are the best structure theorem that are known at present? All I know is a result that every finite commutative ring is a direct product of local commutative rings (this is correct, right?) in some paper which computes the size of the general linear group over that ring.

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 $p$. 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 After that, the nonzero monomials may be linearly dependent (and never mind that $R$ might be more complicated than its associated graded). 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 NPhard to determine when two such trilinear forms are equivalent. NPhardness 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. The situation is similar to noncommutative $p$groups, which are also wild and will never be classified. In both cases, certain classes have a nice structure. It is also interesting to make estimates for how many there are. Note: Corrected per comment. 


This is a very interesting question related to the Hilbert scheme $Hilb^n(\mathbb A^d)$ classifying $n$ points in affine space $\mathbb A^d$. I don't think there is a classification but there is an estimate for the number of commutative rings of order $\leq N$. It is $$exp[\frac{2}{27} \frac{log(N)^3}{(log 2)^2} \; +O(log(N)^{\frac {8}{3}})] \quad for N\to \infty $$ The proof of this result due to Bjorn Poonen and of many related interesting theorems is in his article You will also find astonishing conjectures in the article like: The fraction of local rings of order $\leq N$ among all commutative rings $A$ of order $\leq N$ tends to 1. Same limit 1 for the fraction of rings "of characteristic 8" in the sense that $8 . 1_A =0$ but $4 .1 _A \neq 0$. 


The characterization of Artinian rings is relevant of course. See also the book "Finite commutative rings and their applications" and this web page. 


As always one should check out the OEIS for questions of this type. In this case see http://oeis.org/A027623 


Assuming that by finite you mean "the underlying set is finite", then the very first thought that comes to mind is that the underlying additive group of your ring has to be finite. I would hope that you are then able to look up or work out what one might be able to say about finite abelian groups, and presumably that might speed you on your way. Maybe I've misunderstood your terminology? 

