Let me start by something classical: extending the classical result for PIDs, by Steinitz's Theorem (1912) all finitely generated modules over Dedekind domains are characterized, see http://en.wikipedia.org/wiki/Dedekind_domains and scroll down.
Beyond Dedekind domains things get complicated, but there is considerable recent work going on. The following paper by Levy and Klingler gives an overview on their expansive (recent) work on this. Very roughly, there is a tame/wild dichotomy (very informally 'wild' is more or less 'hopeless') and they give an (essentially) complete answer for those noetherian rings were the answer is not 'wild'; the key-word here is Dedekind-like.
The 'essential' refers to the fact that, as they point out, there are or at least might be some exceptions in some characteristic two cases.
I am convinced that I once saw some subsequent work on this particular cases (though I do not remember whether it was partial or complete), yet unfortunately I am unable to find it right now.