[jauntbox]

You're describing the "Fast" Fourier Transform, which is a specific algorithm for efficiently calculating the Discrete Fourier Transform of a signal.

The plain ol' Fourier Transform is also a bit different than what's described in this blog post. Fourier Transforms can be thought of as an extension of the Fourier Series described there. The Fourier Series shown there all have not only finitely many terms, but also a finite frequency spacing between successive terms (eg. 1Hz, 2Hz, 3Hz, ...). Fourier Transforms build up signals from _all_ frequencies, so are expressed as an integral over frequency components instead of a sum, even if it's an infinite sum.