Overview
There are four main classes of amplifiers; A, A/B, B and D.
Class A amplifiers gives the least distortion, but are terribly inefficient. They are rarely used in commercial amplifiers because they produce too much heat and give too little output power for the effort. In class A the output transistor(s) are always conducting, even at idle with no signal. The power draw of a class A amplifier is roughly constant and they are most efficient at full-output. If the load ends up drawing too much current the amplifier can wind up leaving class A (one transistor switches off in a push-pull, for example) during a cycle.
Class A/B is what most commercial amplifiers are. They are much more efficient, but do produce more distortion. A class A/B amplifier is formed when a class B style output stage is biased so that around the crossover point both transistors are conducting. This yields more distortion than either a proper class A or class B amplifier, however the bias point is much less critical.
Class B amplifiers have two transistors, one per supply rail. In properly-biased class B, only one conducts at a given time, but there is always one conducting. Much of the bad name class B has is due to amplifiers actually being underbiased into class C where there's a portion of the cycle around the crossover point where neither is conducting. Bias is super-critical in a class B amplifier and it's usually easier just to overbias into class A/B. Douglas Self has [an excellent analysis] of distortion mechanisms in class B amplifiers on [his site].
Furthermore, Class B proper biasing may be made very difficult to achieve due to thermal stability issues (especially proper thermal coupling and tracking between the biasing circuit and the output power devices).
Class C amplifiers are useless for audio. They are used in RF applications where the harmonics can be filtered out. Nonetheless, the so called "Class G" is just the plain combining of a normal Class AB output stage with a Class C "booster" enabled to operate only if high power peak are required by the load. If properly designed they performances are equivalent to that obtained by normal class AB amplifiers.
Class D amplifiers are a rather new phenomenon in the hifi world. They are extremely efficient (80%) and can give a very good result. They use pulse width modulation to amplify the signal; this lets them use the output transistors in switch mode where they're most efficient and dissipate the least power. Originally only for subwoofers, newer designs have since surfaced making this technology capable of sounding very good as a full-range amplifier.
NOTE - In the november 2006, Douglas Self has proposed, on Electronic World magazine, the so called "Class XD"amplifier, a mixed class power amplifier where an output stage equipped with *three* power devices work as pure class A amplifier for low output currents and as a normal class AB amplifier when higher output current (in excess of 1 Ampere) is required by the load. The concept is not entirely new or unsual but this appear to be his first commercial application.
Links
[Wikipedia's page] - Has some more info on classes not mentioned here (with pictures)
[Tripath] - Improved Class D ("Class T") topology
[Class XD] - In the essence, the same technical content of the articled appeared on "Electronic World", november, 2006
