What is the DFT and how does it relate to FFT?

The DFT (Discrete Fourier Transform) computes the frequency content of a finite-length discrete signal: X[k] = sum(x[n]*e^(-j2*pi*k*n/N)). It transforms N time samples to N frequency samples. The FFT (Fast Fourier Transform) is an efficient algorithm for computing DFT, reducing complexity from O(N^2) to O(N*log N). Common FFT algorithms: Radix-2 (N is power of 2), Radix-4, and split-radix. FFT enables practical spectral analysis, fast convolution (multiply in frequency domain), and many signal processing applications.