If, like me, you struggled to understand the Fourier Transformation when you first learned about it, this succinct one-sentence colour-coded explanation from Stuart Riffle probably comes several years too late:
Stuart provides a more detailed explanation here. This is the formula for the Discrete Fourier Transform, which converts sampled signals (like a digital sound recording) into the frequency domain (what tones are represented in the sound, and at what energies?). It's the mathematical engine behind a lot of the technology you use today, including mp3 files, file compression, and even how your old AM radio stays in tune.
The daunting formula involves imaginary numbers and complex summations, but Stuart's idea is simple. Imagine an enormous speaker, mounted on a pole, playing a repeating sound. The speaker is so large, you can see the cone move back and forth with the sound. Mark a point on the cone, and now rotate the pole. Trace the point from an above-ground view, if the resulting squiggly curve is off-center, then there is frequency corresponding the pole's rotational frequency is represented in the sound. This animated illustration (click to see it in action) illustrates the process:
The upper signal is make up of three frequencies ("notes"), but only the bottom-right squiggle is generated by a rotational frequency matching one of the component frequencies of the signal.
By the way, no-one uses that formula to actually calculate the Discrete Fourier Transform — use the Fast Fourier Transform instead, as implemented by the fft function in R. As the name suggests, it's much faster.
AltDevBlog: Understanding the Fourier Transform (note: link no longer active)