First of all, a Fourier series is a simpler notion, so we'll start with that. If $f(x)$ is a nice(ish) function that is periodic with period $2L$ (i.e. $f(2L)=f(0)$), then we can write:

$f(x)=(1/2)a_0+\sum _{n=1}^{\infty }(a_n\cos (n\pi x/L)+b_n\sin (n\pi x/L)$

for some coefficients $a_n$ and $b_n$. In fact, we can work out these coefficients from $f$:

$a_n=(1/L){\int }_0^{2L}f(x)\cos (n\pi x/L)\mathrm{dx}$

$b_n=(1/L){\int }_0^{2L}f(x)\sin (n\pi x/L)\mathrm{dx}$

(I might be out by a constant factor here, I'm not sure).

Why would anyone do this? Well, there are various reasons. Firstly, you can single out frequency components by finding the Fourier series, and use this to find the effect of high pass filters on sound signals for instance (a high pass filter removes high frequencies from a sound signal, which can be used for (e.g.) removing hiss from tape). It's also useful in physics quite a lot for solving partial differential equations with boundary conditions (e.g. Laplace's equation).

A Fourier transformation (FT) is a related notion. The Fourier transformation of a function $f(x)$ is $F(\omega )=(1/\sqrt{2\pi }){\int }_{-\infty }^{-\infty }{e}^{-i\omega x}f(x)\mathrm{dx}$

The inverse Fourier transform of $F(\omega )$ is

$f(x)=(1/\sqrt{2\pi }){\int }_{-\infty }^{\infty }{e}^{i\omega x}F(\omega )d\omega$

The Fourier inversion theorem states (roughly) that the inverse Fourier transform of the Fourier transform of a function is the original function. Actually, it is not quite the original function, it is the original function where it is continuous, and it is the midpoint of the upper and lower values of the function at a discontinuity. For periodic functions, the inverse FT of an FT simplifies to the Fourier series above. The inverse FT of the FT is sometimes called the Fourier representation of the function.

Fourier transforms have various properties which can be deduced from the definition, one important one is the following. If $g(x)=f\text{'}(x)$ the derivative of $f$, then the FT of $g$ is $G(\omega )=(i\omega )F(\omega )$. This can help us to solve differential equations. For instance, if we want to solve:

$y\text{'}\text{'}(x)+ay\text{'}(x)+by(x)=f(x)$

Then we take the FT and get

$(-{\omega }^2)Y(\omega )+ai\omega Y(\omega )+bY(\omega )=F(\omega )$

Which we can easily solve for $Y(\omega )$. Now we just take the inverse FT of $Y(\omega )$ and we get the original function $y(x)$ which satisfies the differential equation.

There are lots of uses of Fourier transforms, in the other discussion for instance (Money) I mentioned that the Fourier transform of position space is momentum space in Quantum Physics. Someone has just made a slightly more in depth post about this.

So there you go, Fourier in a nutshell. I've just finished revising it for my exams as it happens, so if anyone notices any big mistakes, PLEASE TELL ME! Did you follow all that?