I'll give an engineering answer.

If you have a time series that you think is the result of a additive collection of periodic function, the Fourier transform will help you determine what the dominant frequencies are.

This is the way guitar tuners work. The perform and FFT on the sound data and pick out the frequency with the greatest power (squares of the real and imaginary parts) and consider that the "note." This is called the fundamental frequency.

There are many other uses, so you might want to add big list as a tag.