Why is one just interested in classifying (the models of) complete theories?
For example, the wikipedia page on stable theories writes:
One goal of classification theory is to divide all complete theories into those whose models can be classified and those whose models are too complicated to classify, and to classify all models in the cases where this can be done.
One could state this research problem also for arbitrary theories: for which theories is it possible to classify the models?
Why does one mainly restrict research for complete theories?
The simple answer is that it's easier: complete theories are simpler to work with (since any two models are elementarily equivalent - this gives us tools like elementary amalgamation, uniqueness of saturated models in the cardinalities where they exist, etc.), and restricting the classification problem to the complete theories reduces the scope of the work we have to do. Of course, even this restricted problem is already quite hard! And it leads to interesting mathematics, much of which works better in the context of a complete theory than outside that context. So most model theorists are happy to stick to complete theories.
Also, there is a sense in which the problem of classifying the models of arbitrary theories reduces to the problem of classifying the models of complete theories. Given an arbitrary theory $T$, and a model $M\models T$, the theory $\text{Th}(M) = \{\varphi\mid M\models \varphi\}$ is a complete theory extending $T$. So if you want to understand all the models of $T$, you can first understand all the completions of $T$, and then understand all the models of each completion $T'$ of $T$.
Another way of saying the same thing: If you want to classify all the models of a theory $T$, a natural (but fairly coarse) invariant to start with is the complete theory of each model. And then we can search for finer invariants just within the models of that complete theory.
I should mention that there is quite a bit of work (much of it by Shelah) on classification theory outside the context of complete first-order theories: e.g. classification theory for universal classes, classification theory over a predicate, infinitary logics, AECs, etc.