I don't know much about the different classes, but I do know that the higher classes, C, D etc run WAY cooler and can put out alot of power but the quality of sound is less than that of a Class A amp. Class C, D+ amps are mainly used for powering subwoofers, mostly in Car audio and in home audio as well. Class A amps have very high sound quality but tend to run very hot and are usually VERY expensive, like the Krell Master Reference amp.
Here is what I've heard, but I honestly don't know and a lot of this is probably wrong. I'm sure someone else will tell us the correct information though.
Class A amps 'bias' the output transistors with a decent amount of current (1-2+ amps) to improve their linearity and reduce distortion. Of course this comes at a cost in efficiency (50% maximum and usually a lot less) and heat - often over 100W idle dissipation.
Class AB amps have a smaller bias (usually a fraction of an amp, I think). Most solid state amps are run Class AB because it's more efficient than Class A but still sounds better than Class B. Don't take my word for it though.
Class B amps have no bias on the transistors.
I believe Class D, F, etc are digital or switching amplifiers. I don't know how they work but they are very efficient and great for subwoofer amplifiers. There are also companies developing these amps for hi-fi applications and the results are getting better.