FFT (fast Fourier transform)
The algorithm that splits a signal into its frequencies in n·log n steps instead of n² — which is what made digital audio processing practical at all.
Computed naively, a discrete Fourier transform (DFT) costs n² multiplications: 1,024 samples means a million operations. In 1965 James Cooley and John Tukey published a divide-and-conquer version that recursively halves the problem and gets it down to roughly ten thousand. Gauss had essentially the same idea around 1805, but wrote it up in Latin in an unpublished note, which limited its reach somewhat.
In audio it is everywhere: the spectrum display on an equalizer, noise reduction, time-stretching, and fast convolution — the only reason a convolution reverb with a multi-second impulse response can run in real time. MIT's FFTW library is still the benchmark everyone else measures against.