TITLE: What is Model Theory
QUESTION [16 upvotes]: I've been reading up on some Model Theory lately. My question is simple. What is the purpose of the theory to begin with? How does it enrich mathematics? A-priori, it doesn't seem like we're doing much, and the theory doesn't seem awfully too far from algebra in many instances. For instance, you could interpret the question of $\pi$ or $e$ being algebraic numbers, as a question of definability in model theory. But what's the significance of doing that? Would this be an exercise in futility, since we would have to fall back to algebraic reasonings anyway? Does model theory provide us with new tools to study such questions?

What's the significance of introducing and coming up with theorems about languages, how does algebra fall short of that?
What's the significance of studying different models of the same theory? It is tempting to say that they are all the "same" (certainly at least up to some equivalence relation).

REPLY [6 votes]: Noah makes a good point.  In fact, the noted representation theorist David Kazhdan wrote in the first chapter (logic) of his notes on Motivic integration wrote:
 On the first glance one should not
distinguish between different models of T, since all the results
which are true in one model of T are true in any other model. One of
main observations of the Model theory says that our decision to
ignore the existence of differences between models is too hasty.
Different models of complete theories are of different flavors and
support different intuitions\footnote{Examples are atomic,
homogenous, and saturated models.}. So an attack on a problem often
starts [with] a choice of an appropriate model. Such an approach lead
to many non-trivial techniques for constructions of models which all
are based on the compactness theorem which is almost the same as the
fundamental existence theorem.
http://www.ma.huji.ac.il/~kazhdan/Notes/motivic/b.pdf
If it is not too self-promoting let me recommend my book. Model Theory and the Philosophy of Mathematical Practice: Formalization without Foundationalism 
by John T. Baldwin (Author)  Cambridge Press 2018.