Classification theory is the study of the conditions under which the models (or, at least, some suitable subclass of the models) of a theory may be explicitly described by reasonable combinatorial invariants (cardinal invariants, some kind of tree, et cetera).
For a complete theory T and cardinal &lambda ≥ ℵ0 define I(T,&lambda) to be the number of isomorphism types of models of T of cardinality λ. In its most basic form, classification theory concerns the computation of I(T,λ).
However, Shelah is dismissive of this point of view:
Though some think this [computing the possible functions I(T,λ) for countable T] was "the problem," I could never make myself excited about it. Still it would be nice to know.and
Some people think this [Vaught's conjecture, namely I(T,ℵ0) < 2ℵ0 ⇒ I(T,ℵ0) ≤ ℵ0 for countable T] is the most important question in model theory as its solution will give us an understanding of countable models which is the most important kind of models. We disagree with all those three statements.