TITLE: (Finite) Classification Theory
QUESTION [13 upvotes]: In the context of asking about the classification of finite simple groups, 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 which apparently is unreadable. My impression is that the notion of "classifiable" in this subject is that of a stable theory, 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?

REPLY [3 votes]: In addition to Andreas' remark on the number of non-isomorphic models, perhaps it's noteworthy to say that many non-structure theorems are aiming at the construction of many models which not only are non-isomorphic, but also are hard to tell apart. This may be achieved by requiring the models to satisfy the same set of sentences of some infinitary logic. By taking infinitary logics into consideration, we may consider a theory T to be classifiable if elementary equivalence in L_(infty, lambda) is a sufficient condition for an isomorphism between a given pair of models of cardinality lambda (this should be found in Shelah's book).