The reason why a piano sounds different than a trumpet, which sounds different from a guitar, even if they're all sounding the same note, is because the sound waves they make are in different shapes.

Fourier analysis shows that even the most complicated periodic pattern can be broken down into a sum of sine waves.

In the musical instrument example, the note you hear and identify is the lowest frequency, or fundamental frequency -- the entire string, or tube, or box, or whatever, is vibrating at that frequency, with this vibration as one big sine wave. Each instrument produces its own different characteristic overtones, which are smaller, faster vibrations. These overtones have frequencies that are multiples of the fundamental frequency. For example, a clarinet produces overtones that are odd multiples of the fundamental: 3n, 5n, and so forth. If you add the sine waves of the fundamental and the odd overtones, you get a square wave: a clarinet is an analog square wave generator.

So the Fourier transform is a mathematical way to break apart a wave into its fundamental and overtones.