A simple explanation
Puting it into terms that everyone can understand; a Class A amplifier is a type of amplifier where the current is always on and sent to the speakers regardless of the musical signal.
You tend to find a warmer sound and higher power consumption as the current is always being converted to heat.
The Bryston BP17 Cubed is an example of a premium Class A preamplifier.
With Class A/B, power is reduced where there is no signal but as the signal demands it power consumption in increased to meet the demands and dynamics of the music.
It offers a more efficient option to Class A and the sound is typically more reined in so far as smoothness. This is probably the most common type of Audiophile amplifier consequently. An example is the Cyrus Mono X300 Signature mono power amps
With Class D, the amplifier is creating pulses related to the original signal, rather than being on all the time, which makes the amplifier very efficient with little heat produced.
As a downside you get high frequency noise created by these pulses which has to be electrically filtered to deal with it.
Class D typically produces a sound that is a bit more pitched up in the high frequencies, although some commentators believe that the sound on the newest amplifiers are similar. I would agree with some of them.
It is typically used in high power applications or inside active powered speakers where cooling might be an issue, for example.
Cyrus’ ONE Cast uses a class D integrated amplifier which sounds very much like a Class A/B amplifier.
For an article about the differences between active and passive crossovers and speakers click here.