Fourier transforms convert complex signals into a series of sinusoid waves. Since it's easy to create a sine wave with a microcontroller, you can perform a Fourier transform on a complex wave and reproduce it with sine waves. A few sine waves could be used to generate the sounds of instruments, and many lossy files like JPEG are compressed using DCT, a form of Fourier transform.
But the math can be a little complicated. I watched a few youtube lectures on Fourier series and they were intolerably slow. Fortunately, I found a site that teaches FFT with interactive flash tutorials - it's easier to understand the concepts when you can see them work, play with them in real-time, and skip past the topics you already understand. I found it pretty useful and thought I'd share it for anyone else who has an interest signals and needs a refresher, the url is www.fourier-series.com.
Fourier - compressed with an algorithm he helped create!