A type of amplitude modulation where the input levels of the carrier and modulation signals are balanced such that the original carrier and modulation frequencies disappear totally from the output, leaving only the sum and difference frequencies. The resulting frequencies are almost guaranteed to not be harmonically related, and ring modulation is often used to simulate the sounds of tuned percussion instruments that produce inharmonic frequency spectra, such as bells and chimes. Ring modulation can be a very difficult effect to control, but it can also produce timbres that are difficult to achieve by any other method of synthesis.
In radio, a circuit involving a ring of diodes and transformers is used to achieve ring modulation. (Confusingly, radio engineers refer to this process as "mixing", although it is totally unlike audio mixing.) This can be done in audio, but the circuit has a very low output level, and it is difficult to prevent distortion. So, analog synthesizers which implement a ring modulator usually do it using a four quadrant voltage controlled amplifier. Radio purists scoff that this isn't a true "balanced" modulator, but if the VCA is calibrated properly, the effect is the same. In contrast, by happy accident of mathematics, it turns out that ring modulation is very easy to compute in the digital domain (just multiply the two signals). So most digital synthesizers include ring modulation as an available function, since it requires little additional software.
Mathematically, ring modulation is just a special case of amplitude modulation, in which the carrier and modulating signal are set equal in amplitude inside the modulator. This is why, with a true ring modulator circuit, it does not matter which signal is routed to which input; they are treated equally inside the modulator circuit.