Class "A" is pretty accuratley described above. You only need 1 transistor in theory but would use more for more power. The bias level (operating voltage on the) transistors keeps them "on" all the time regardless of input signal. This is very innefficient and generates a lot of heat. It achieves a max. efficiency of 25%. This means for 100 watts output, you need at least 400 watts out of your wall. Gryphon amps and the Pass Aleph use this method. Many manufacturers, such as the newer Krell amps, say that they have "class A" but they are fibbing, what they do is have a "sliding bias". So at low input levels you have a low bias which improves efficiency, and then when the volume is turned up, you up the bias. This would be good if you had a buffer and could therefore predict the input. Your amp is not class A unless it is very, very hot with no input.
Class "B" is academic and not used for audio amps, this class will only turn "on" a transistor if there is either a + or - input to it. However, near the zero point it will be "off" because of diode voltage drop charactertistic of the transistor. You need a + and a - transistor for class B (only 1 transistor needed for class A) One transistor handles the + phase of the signal, and if you need - amplification, which you do, the other handles that. But in the crossover region, (.7 v or less closer to zero for a BJT xistor) you will get nothing, called crossover distortion. Humans are very sensitive to crossover distortion and it sounds very bad, painfully bad, so class B amps are NEVER EVER used. As an aside, I know a study that was done where class B was used as a video amp to see if it would cause visual pain but none resulted.
Class AB- the most common amplifier out there. This is a class B amplifier with the transistors biased to at least the crossover point, say .7 volts. This means there are at least two transistors, 1 for + and 1 for - that are biased "on" for very low voltages. They are efficient and run in class "A" for less than .7 volt inputs, but class B for all others, which is ok, since the crossover distortion is not around anymore. The problem with AB is that you need to closely match your + and - transistors or they will work differently on the same signal causing mucho trouble.
Why bias it at .7 when you can go higher? A class AB amp might be "highly class A" which means that instead of biasing it for the crossover distortion (.7 volts say) it is biased for like 4 volts, in both the + and - transistors. This means that its operating in class A for that part and "B" beyond, so its less efficient class A for low level inputs but more efficient when you crank it up. The newer Pass amps use this.