Intuitive

The Fourier transform is a mathematical tool that allows us to split a function into its fundamental constituents.

Concrete

• The best short introduction is An Interactive Guide To The Fourier Transform by Kalid Azad and the best longer introduction is the book "Who Is Fourier? A Mathematical Adventure"
• Fourier Analysis by Körner is a good book
• A nice description of how the Fourier transformation is used in quantum mechanics, can be found at page 246ff "The Emperor's New Mind" by R. Penrose.

Abstract

Why is it interesting?

The Fourier transform is an essential tool that, for example, makes the music format MP3 possible. With the help of the Fourier transform any given music can be split into the frequencies that it is built of. Then, we can remove those that can't be heard by humans and get an equally sounding, but much smaller music file.

Moreover, the Fourier transform is essential in Quantum Field Theory, where we usually expand any given field configuration in terms of it's Fourier modes and then start analyzing it.

One can Fourier transform anything—often meaningfully.

John Tukey