The class definitions come from the way the circuits are "biased". It has to do with how much power the circuit uses when no signal is present, and therefore efficiency.
Class A draws 50% power with no signal, so it can only be 50% effiecient best case. The upside is it has the absolute lowest distortion.
Class B has a theoretical effieciency of 78%. But has lots of distortion. Used only in portable voice amps (like bullhorns) where battery life is more important than sound quality.
Class AB is somewhere in between. It's the most popular configuration. They run maybe 70% efficient (not including their internal power supply) and can have very low distortion.
Class D uses digital switching technology instead of analog circuits. The signal is chopped up into narrow pulses and fed into switching transistors that are either on or off. Then it's fed back through a lowpass filter to "reconstruct" the signal. They have excellent effieciency at the expense of having zero high frequency response and somewhat higher distortion. Perfect for sub amps.
Class T is a new technology where I believe they modulate the power supply instead of the output stage (not sure about that). Supposedly gives high effieciency without the dawbacks of Class D. They cost bucks tho.