The Fourier transform is a way to analyze a waveform and determine the correct combination of pure sine waves and cosine waves that can be combined to re-create the original wave. Knowing the composition of an arbitrary wave in this way can enable all sorts of analysis and processing, including advanced signal processing.

The problem is that it's highly compute intensive, involving a sequence of integrals and so forth. In most cases, there are other analytic methods that are more practical than Fourier transform because of the work required.

In 1964, the Fast Fourier Transform was invented. It's basically a computational "trick" that replaces all those integrals with simple shifts and adds and subtracts. All of a sudden it became practical to use Fourier transforms in all sorts of ways that weren't practical before.

That video linked by /u/croninsiglos looks really good, off to watch it now.