Fourier series are useful (and sometimes essential) for solving/understanding many problems involving periodic functions on $\mathbb{R}$ or, equivalently, functions $f$ on $[a,b]$ such that $f(a)=f(b)$. I was going to say almost all problems, but that's probably an exaggeration. Of course it helps if the problem is linear, and the properties of the functions you're considering can be easily expressed in terms of the Fourier coefficients -- but even then these restrictions are not always essential.
e.g. the Heat Equation on a (physical) ring, where periodicity is assured by the shape of the space; (Willie Wong already mentioned this in the comments).
My favourite one: proving the Isoperimetric Inequality, that the circle has the largest area of all piecewise $C^1$ curves with given perimeter;
The functional equation for the Riemann Zeta Function $\zeta$: one proof involves the Fourier expansion of the sawtooth function $x - [x]$, which I think I saw in E. C. Titchmarsh's old book The Theory of the Riemann Zeta Function (although I'm sure many other books will give it also).
I think it was either Hardy or Littlewood (or maybe both?!) who said that a periodic function should always be expanded as a Fourier series; if you always follow this rule then it'll solve a lot of problems automatically!
Although one should be cautious; "if the only tool you have is a hammer, then everything looks like a nail"...
