What is 'spectrum' anyway? How can multiple frequencies exist in one signal?

Oric and Joe,
the Arduino lib link is {Reference -> libraries -> Contributed Libraries -> Audio and Waveforms. Then look in FFT.
Open Music labs has done some great work. The interesting source code is in assembler, because the standard libraries are too slow.

If you look in the one of his links to http://www.alwayslearn.com/, this is the other "learning link" I was referring to.

Another useful page is DSP Guru. Just google it.