A Fourier transform takes a function f:A->X, and produces another function g:B->X, such that g describes the "coefficients" of a family of sinusoidal that, when added together, produce f.
g(b) is the "coefficient" in front of the sinusoidal h(x) := e2 pi i b x
. (Yes, ex
is a sinusoidal -- in particular, e2 pi i x
How we add them up depends on our B. Basically, we define a measure on our B, and then integrate with respect to that measure. (a measure is a way of weighting the elements. A counting measure says each element has weight 1. A measure on a continuum (like the real line, or the interval [0,1]) tends to say any one element has weight 0, while intervals have non-zero weight.)
The wonder of the Fourier Transform is the symmetry between extracting these coefficients, and reconstructing the original signal. How that works is going to be beyond what I could put in a post.
(anyone else want to try?) But the basic idea remains: you are building a map from the space of sinusoidal (in this case, sinusoidals with a complex index!) to coefficients on said sinusoidals. You then add up each of those (integrate them together, because adding up an uncountable number of functions doesn't work well without integrating them) to generate your original function.
As another example, FFT uses a discrete family of sinusoidal indexed by a discrete set B. This changes how we add them up, and restricts what set of functions we can represent. In this case, the "sum of sinusodials" is more obvious, because instead of this strange looking integral, we get an actual sum of functions.