discrete Fourier transform (DFT)

The discrete Fourier transform takes a finite block of N samples and tells you 'how much of each pure tone is hiding inside it.' Feed it a chord and it hands back the individual notes; feed it the rumble of a bridge and it reveals the resonant frequencies that might shake it apart. It is the workhorse that lets a computer move from the time domain (what the signal looks like moment to moment) to the frequency domain (what tones it is built from).

Mechanically it correlates your N samples against N reference sinusoids and reports a complex number for each — magnitude (how strong that frequency is) and phase (its timing). A crucial catch: the DFT assumes your block repeats forever, so a tone that doesn't fit a whole number of cycles inside the window smears its energy across neighbouring bins, an effect called spectral leakage that windowing is invented to tame.