If you want to see more math behind a Fourier Transform, read my DFT notes.
If you want to see me write a really crappy FFT, read my FFT notes.
If you don't know what a Hann window is, read my windowing notes.

Sonogram

The problem with a Fourier Transform is that while we can find the amplitudes of the different frequencies that are present, we can't know at what exact time they occur. One way of visualising approximately how amplitudes change through time is by taking lots of overlapping sections of the signal we're analyzing, transforming those sections, and graphing the results along a time axis. The result is called a sonogram or a voice-print.

This is a plain sonogram of nF's guitar sample:

And here's one using the Hann window to taper sections of the signal before transforming them:

The Hann window reduces the spectral leakage a lot at the expense of lowering the resolution (the peaks become fatter). As always, you can grab the code. (sonogram.c)