Ah, uhm, eh. Here is an oversimplified description written in plain english for which I'm probably going to be lynched by the techie wonks (mercy!). For power amps:
Class A: the working point of the electronics (full signal) is such that all power is dissipated in the amplifier itself, unless sound is produced in the speakers (or phones). If sound is produced, the electronics will dissipate the difference between the nominal power (plus losses) and the power delivered to the speakers. I.e. the higher the volume, the less heat the amplifier generates. Achievable (not guaranteed!) advantages: low distortion with relatively little feedback (that's a good thing), hence good transient response with low TID, *great* sound quality, no cross over distortion by design. Disadvantages: except for small power output, the amp is in fact a very expensive heating appliance most of the time; wasteful of energy; expensive.
Class B: in general, not an amp you want to use if you are frequenting this forum
because of cross-over distortion.
Class AB: like class A for low volume in the speakers, but for higher power/volume, the amp delivers to the speaker all the power, except for losses which it still dissipates. This is a popular desing as it offers the best bang for the quality watt when well designed. Achevable advantages: higher power rating, very good sound quality, usually cheaper to build than class A for same power, more efficient (not a heating appliance). Disadvantages: uh, it's not class A (but still can be very, very good).
Class C: see class B, only an awful lot worse. Efficient use of electricity. Lots of power for relatively little cost. Usually popular in RF (radio frequency) power applications.
Class D: switching amplifier. Ych, ugh, bleah! Very efficient, not for audio hi-fi.