FM synthesis, short for frequency modulation synthesis, was discovered by John Chowning at Stanford University in the sixties. It was patented by Stanford University, and licensed to Yamaha, who used it to make their famous DX range of synthesisers. The DX-1 was the first synthesiser to employ FM synthesis, while the DX-7 was by far the most popular.
Essentially all you need to do is feed the output of an oscillator into the input of another oscillator, so that the frequency of the second one is determined by the output of the first. It works when both frequencies are of an audible rate (roughly between 20Hz and 20kHz). This means it's possible to use FM synthesis on analogue synthesisers, but you'd quickly run out of components when trying to make complex sounds and voltage controlled oscillators probably wouldn't be stable enough.
Usually the sounds produced by FM synthesis are not very musical sounding, as the frequencies of the side bands produced aren't multiples of the fundamental harmonic (ie the note you're actually playing). You need to keep the carrier to modulator ratio simple in order to make pleasant timbres.
Incidentally, the Yamaha chip inside the Sega Genesis (aka Mega Drive) uses FM synthesis to make most of the sounds. Yuzo Koshiro's Streets of Rage soundtrack is an example of excellent use of this synthesis method, although it uses a few PCM samples too.