There is some classification of finite commutative local rings. For example how many not isomorphic finite local rings with the same order $p^k$ and the same residue field $\mathbb F\_p$ exist?



Finite local rings
Miguel, I S Cohen's structure theorems on complete local rings answers (more or less) your question, since a finite local ring is complete. Your ring $R$ must be a quotient of a ring of power series $k[[x\_1,\dots, x\_n]]$ where $k$ is either a finite field or the ring of Witt vectors over a finite field. You can take $n=m$ to be the minimal number of generators of the maximal ideal $\mathfrak{m}=(y\_1,\dots,y\_m)\subset R$, or $n=m-1$ if $R$ does not contain the residue field. 


The answer to your specific question looks like more complicated in general, but you can try to play with the theorem to get an answer. For instance, the only examples with $p^2$ elements are simply $\mathbb{F}\_p[t]/(t^2)$, $\mathbb{Z}/(p^2)$, and $\mathbb{F}\_{p^2}$.


I. S. Cohen
On the Structure and Ideal Theory of Complete Local Rings
Transactions of the American Mathematical Society, Vol. 59, No. 1 (Jan., 1946), pp. 54-106


