Real world application of Fourier series
It turns out that (almost) any kind of a wave can be written as a sum of sines and cosines. So for example, if I was to record your voice for one second saying something, I can find its fourier series which may look something like this for example
$$\textrm{voice} = \sin(x)+\frac{1}{10}\sin(2x)+\frac{1}{100}\sin(3x)+\cdots$$
and this interactive module shows you how when you add sines and/or cosines the graph of cosines and sines becomes closer and closer to the original graph we are trying to approximate.
The really cool thing about fourier series is that first, almost any kind of a wave can be approximated. Second, when fourier series converge, they converge very fast.
So one of many many applications is compression. Everyone's favorite MP3 format uses this for audio compression. You take a sound, expand its fourier series. It'll most likely be an infinite series BUT it converges so fast that taking the first few terms is enough to reproduce the original sound. The rest of the terms can be ignored because they add so little that a human ear can likely tell no difference. So I just save the first few terms and then use them to reproduce the sound whenever I want to listen to it and it takes much less memory.
JPEG for pictures is the same idea.
I can say about these applications.
Signal Processing. It may be the best application of Fourier analysis.
Approximation Theory. We use Fourier series to write a function as a trigonometric polynomial.
Control Theory. The Fourier series of functions in the differential equation often gives some prediction about the behavior of the solution of differential equation. They are useful to find out the dynamics of the solution.
Partial Differential equation. We use it to solve higher order partial differential equations by the method of separation of variables.
For a very specific example: One of our undergraduate students was taking data generated by a person running on a force plate. Since force exerted on your feet from running is for the most part periodic, she fit the data with a curve using Fourier analysis. The work that followed can be used to help develop better running shoes.