What are modular forms used for? I have seen the definition of a modular form, but it seems obscure to me. I get the impression that if I were to read a lot about them, eventually I would see how they can be used. I am curious about the ways in which modular forms are applied. How are they used? What are some important theorems of intrinsic interest that can be (relatively easily) obtained by using them? Are there any that I should look at in particular?
 A: One of the simplest applications (and quickest to get to) is to representation numbers of quadratic forms.  E.g. Jacobi's formula, that the number
of ways of writing a natural number $n$ as the sum of four squares is equal
to $8 \sum_{d|n, 4 \not\mid d} d$, was originally proved using modular forms,
and I think this is still the most versatile method of proof.
For more general quadratic forms, one can't necessarily get as precise formulas
(so-called cuspforms introduce error terms which don't admit explicit formulas),
but one gets approximations (and the Ramanujan--Petersson conjecture on growth
of Fourier coeffs. of cuspforms plays a role in bounding the error  terms coming
from cuspforms).
Some of this (although not Jacobi's formula itself) can be found in Serre's Course in arithmetic, which is the nicest treatment for a beginner.

There are also the applications to the theory of elliptic curves (and then to FLT) mentioned in the comments.  For example, the best results in the direction of BSD (such as Gross--Zagier, or Kato's results) rely on the connection between modular forms and elliptic curves.
