Here is a brief look at the classes of solid-state audio amplifiers. These classifications do not include tube amps (valves, for you Brits).
Class A Amplifiers
The output device(s) operate over the full 360° of the waveform. The output devices never turn off. Efficiency of the output stage is very low, with either 12.5%, 25%, or 50% theoretical maximum efficiency.
Class A amplifiers are highly touted for their linearity (low distortion). Most often this is true, but sometimes it isn’t. A class A amplifier can be as simple as a single transistor and a few biasing components, or as complex as any other. One thing that is always true about class A is that it is horribly inefficient, since the full output current capability of the amplifier is always flowing in the amplifier. Current that is not delivered to the load is wasted as heat. This fact alone usually relegates class A designs to lower output levels.
Still, a well-designed class A amplifier can put up some very impressive numbers. And a poorly designed (or should I say conceived?) one will do no better than a tube amp. (Oops, did I say that?)
Class B Amplifiers
The output device operates for 180° of the waveform. A complementary pair of output devices is used to deliver the full 360° of the waveform. Efficiency is much improved over Class A operation, since the quiescent current is very low compared to Class A. Most people call Class B amps “Class AB” (a misnomer, in my opinion) due to the presence of a small bias voltage. If the bias voltage were not present, then the output devices would deliver power to the load for less than 180° of the waveform and you would have crossover distortion and technically you are now in Class C operation.
Class B amplifiers make up the bulk of commercial and professional power amplifiers. Although in my opinion it is technically incorrect to call them Class AB amplifiers, most people do (even engineers) and most people will say, “Class B? No, class B amps will have crossover distortion and aren’t any good for Hi-Fi.” This is again one of those nomenclature things that has been around so long that it’s almost of no use trying to convince people otherwise. In fact the term “Class B” has such a negative connotation that the sales and marketing departments would probably still refuse to use it even if it were technically correct. Each output device in a correctly designed Class B will deliver current to the load for exactly 180° of a sine wave cycle. A pair of output devices, precisely timed, take turns delivering current to the load. If it does not deliver current to the load for a full 180° the result will be crossover distortion which is a very clearly audible form of distortion due the presence of high-order harmonics. A well designed class B amplifier will have no audible crossover distortion. This is done by not allowing the output transistors to fully turn off at the crossover point (zero volts on the output). A small bias voltage on the output devices allows a small amount of quiescent current (aka bias current) to flow in the output devices even though there is no output voltage (the complementary device is sinking the current), and therefore no output current to the loudspeaker.
To explain it any further requires a picture… so let me know if this is of any interest and I will try to get it done.
Class AB Amplifiers
Most Hi-Fi amplifiers are marketed as Class AB amplifiers, but are truly Class B. Please read the aforementioned stuff in the Class B section for my explanation.
An otherwise Class B amplifier that is heavily biased can also be termed a Class AB amplifier. It will run in class A for a (usually) small part of the output power then revert to Class B (sort of) operation at higher power levels. The other case of Class AB operation is when a class A amplifier drives a heavy (low impedance) load slips into class B operation when it can’t deliver the required current in class A mode. This type of operation could be intentional or unintentional (or perhaps not designed for). One curious type of amplifier is a “trimodal” design that can operate in Class A, Class B or Class AB at the flick of a switch.
Class AB (as described here) will actually have more distortion than an optimally designed (and optimally biased) class B amplifier. Not enough bias voltage causes crossover distortion, but too much can cause distortion too (though it is not as readily audible), and you get to say “2W of Pure Class A” in the marketing copy. Whoopdy-doo.
The technical case used for calling a “class B” amplifier a “class AB” amplifier is this: at idle the output devices aren’t really “off” but conduct a small quiescent current so it’s Class AB. Whatever. 99% of the people agree with you. 
Maybe it's just bias. (pun intended)
...to be continued