Multiplication in the time domain is the same as convolution in the frequency domain. Thus, in the basic case where two sine waves of frequencies f1 and f2 (f2>f1) are multiplied, two new sine waves are created, with one at f1+f2 and the other at f2-f1.
Interesting harmonics can be generated by carefully selecting and changing the frequency of the two input waveforms.
On the C64 SID chip, ring modulation multiplies a triangle wave with a square wave. A ring modulator was a common feature on early modular Moog synthesizers. The ring modulator went out of fashion with the advent of all-in-one synthesizers sampled-based synthesizers, but has returned as a feature in digital modelling and software synthesizers.