TITLE: Where are the model theory concepts from?
QUESTION [11 upvotes]: Look at the following definition.
Definition. Let $\kappa$ be an infinite cardinal. A theory $T$ is called $\kappa$-stable if for all model $M\models T$ and all $A\subset M$ with $|A|\leq \kappa$ we have $|S_n^M(A)|\leq \kappa$.
A theory $T$ is called stable if it is $\kappa$-stable for some infinite cardinal $\kappa$.
I am beginner in model theory and so my questions might be stupid. When you read a basic textbook in math you see that the algebraic , analysis, geometric, topological and .... definitions/concepts  are natural. For example algebraic concepts, group, ring, field, module, Galois theory and ... have a clear natural roots. Also the notion of continuity in analysis is a natural (in my sense!) concept. The idea behind topology is also natural. But the model theoretic notions are usually not concrete for me at all! For example one of the main portions in model theory is stability theory (which is a part of Shelah classification) in which you need to count the number of types. I would like to know: where is the idea of stability (counting the number of types) come from?
Any reference would be appreciated.

REPLY [6 votes]: Just to add on one point to why the classification program is natural:
One can interpret the model theorist's approach to studying a structure $M$ as assigning to $M$ its ``logical invariances'' which is just a fancy way to refer to the theory $T$ of $M$.
It is therefore natural trying to understand how powerful these logical invariances are, that is, how far is $T$ from characterizing its models up to isomorphism completely.
It turn out by Löwenheim–Skolem theorem that the only case where $T$ has absolute power is absolutely boring:  $T$ has a single finite model up to isomorphism. So the first non-boring  situation where $T$ is powerful is when $T$ has few models in certain infinite cardinality. Stability is a slightly different expression of the idea that $T$ is powerful. Here $T$ has few types over a small parameter sets instead of few models. As Noah pointed out, these two notions of power are closely related.
The primary insight of Morley and later Shelah, in my opinion, is the following: When $T$ is powerful in a certain way, this is due to the fact that models of $T$ are equipped with some kind of special algebraic features (dimension, indepence relation,...) and do not encode complicated combinatorial patte 
From this point of view, one can see the various definitions of stability as equating being powerful (having few types) with having algebraic features (local notion of dimension, independent relation with special properties) and with non encoding complicated combinatorial patterns (in this case an ordering). One can also think of Morley Theorem as a corollary of this phenomenon: These features persist through other cardinality which leads to the theory also has few models in other cardinality as well.