In the context of asking about [the classification of finite simple groups][1], the question arose: what exactly is meant by a "classification"?

Perhaps unsurprisingly, there is in fact a whole branch of model theory called classification theory, and an epynomous book by [Shelah][2] which apparently is unreadable. My impression is that the notion of "classifiable" in this subject is that of a [_stable theory_][3], but probably there are other notions I'm not aware of.  Unfortunately, this notion seems best suited for talking about infinite models of a theory.  Moreover, it's not clear to me that this definition gives a complete answer to my question: 

What does it mean to say that the models of a theory (with some cardinality limit, particularly the case where the models are required to be finite) admit a classification?
 

  [1]: https://mathoverflow.net/questions/38161/heuristic-argument-that-finite-simple-groups-ought-to-be-classifiable
  [2]: http://books.google.com/books?id=5pwf8DGNHckC&lpg=PP1&ots=OIuZJnCKmO&dq=shelah%2520classification%2520theory&pg=PP1#v=onepage&q&f=false
  [3]: http://en.wikipedia.org/wiki/Stable_theory