The Fourier Transform is used in a wide range of disciplines and activities - it can even save your life! Find out how in Plus Magazine's two articles on the mathematics of tomography (the second article discusses the mathematics behind tomography).
For the musically inclined you can find out how the Fourier transform can lead to a career in audio software engineering.
For a short history of the Fourier Transform see this Guardian article.
If you're confused about what the Fourier Transform actually is - Stack Overflow has the answer.
The Fourier Transform can be applied to many things:
One of the reasons it's so powerful is that it breaks down any periodic function into a combination of sines and cosines. You can also do this for orbits with Ptolemy's system of epicycles and deferents: