Many of us have seen Fourier analysis before and may note that it can seem complicated at first glance, but it is comforting to realize that it is well suited to computational solutions. This is the purpose of Discrete Fourier Transforms (DFT), which allow us to reap the virtues of Fourier analysis without the complications of the more analytical and less practical approach seen in math classes. This post will briefly discuss this topic and more specifically Fast Fourier Transforms (FFTs), which is some algorithm used to find the Discrete Fourier Transform.
In general, Fourier analysis is heavily emphasized because it has so many real-world applications. Fourier analysis, at its essence, is breaking a function into components based on sinusoidal functions of different frequencies. I’ve personally found it very useful in analyzing physical behavior especially described by differential equations, which are aptly solved computationally when only numerical solutions can be found. Fourier analysis has applications in several topics in discrete mathematics, many different areas of physics, and even finance and statistics.
As a basic description of the DFT, a signal or function is sampled in order to transform it into the frequency domain, and it can be done relatively easily computationally because its input is a discrete series of numbers. All of the continuous components in normal Fourier analysis go to discrete sums of finite numbers of turns. FFT compute this with specific algorithms, the most commonly used of which is the Cooley-Tukey algorithm. A simplified FFT can be broken down into two parts: finding the discrete series equations and formulating matrices. First, the period of the signal is determined and a number (multiple of two) of samples is taken over the period. The continuous version is defined by (f(t) = ) a sum from negative infinity to infinity of a constant cn*exp(i*wn*time). The frequency wn is defined by 2πn/T. This is still valid for discrete except n is discrete, and the time T is the specific period mentioned above. The summation can be transformed into a closed form, and the integral used to find the constants can also be transformed to a discrete summation. The new version will be something like a summation from zero to one less than the number of samples(2N-1) of cn*exp(i*wn*specific time). The matrix is formulated by Mnk = exp(i*π*n*k/N). This matrix can be used easily to represent the discrete version and is easily manipulated computationally to find the constants necessary.
This discrete analysis can then be used to numerically find solutions that would be impossible to do otherwise. This can then be used for digital signal processing and solving differential equations. This is just an overview, and the sources below can be consulted for a more thorough understanding of the subject.

Sources:
http://mathworld.wolfram.com/DiscreteFourierTransform.html
http://mathworld.wolfram.com/FastFourierTransform.html
http://local.wasp.uwa.edu.au/~pbourke/other/dft/
http://en.wikipedia.org/wiki/Discrete_Fourier_transform
http://en.wikipedia.org/wiki/Fast_Fourier_transform