Amplitude modulation, or AM, is a variation in the amplitude of a signal, according to the instantaneous value of a modulating signal. (Borrowing from radio terminology, the signal being amplitude modulated is usually referred to as the "carrier".) When the modulator is of subsonic frequency, the result is a slow or rapid variation in the volume level of the carrier signal which is referred to as tremolo. When the modulation is of audio frequency, the result is a composite waveform containing the original carrier, and new waveforms at sum and difference of the frequency of the carrier and the frequency of the modulator. (This assumes that the carrier and modulator are both sine waves; if not, then each harmonic present in either signal acts like a separate carrier or modulator.) These sum and diffferences frequencies are often not harmonically related to the carrier, and the results can be bizarre, particularly when complex waveforms containing many harmonics are used. Compare with frequency modulation. For some reason, audio-frequency AM is seldom implemented on commercial synths, although it is easy to do with a modular synthesizer.
Caution! Math ahead...
A basic mathematical equation for amplitude modulation is:
A = (IG + M) * C
where C is the carrier signal, M is the modulating signal, IG is an initial gain for the carrier (determines the relative modulation ratio at a given modulation amplitude), and A is the amplitude-modulated output signal. Ring modulation is simply a special case of this; setting the IG to zero simplifies the equation to:
A = M * C
which is ring modulation. This shows why, with a ring modulator, it is necessary for signal to be present at both inputs; if either M or C is zero, the result is zero. Also, the commutivity of multiplication explains how the carrier and modulation signals can be switched to the opposite inputs, and the result will be the same.