We all have heard different guitar amplifiers being referred to as "Class A" or "Class A/B" etc.

What does that mean to us?

What sets different amplifier classes apart from each other is how the output tubes are conducting current. A Class A amplifier's output tubes are conducting current at all times, even when no sound is going through them. This means that the tubes are working full on no matter what is happening. This has a few tell-tale trademarks. First off, this obviously means that the tubes are getting quite a workout. Therefore these amps tend to run hot. This is one good reason why you should never put something on top of your Vox AC30 which covers the air vents, unless it's a skillet and you want some fried eggs. These amps are very inefficient when you're looking at the amount of current that it needs to amplify compared to the amount of volume they put out. To have more volume, you need more tubes, and they'll all be running full bore all the time. It might take 5 times as many output tubes to give you the same amount of power that you could get with a more efficient circuit.

Tonally, these amps can give a chimey tone that gets a very sweet sounding distortion when pushed to it's limits. Of course the speakers and the rest of the circuit have a lot to do with this as well, but because of the way that the output tubes distort (which is without crossover distortion that you get with other classes) you will get a unique tone. The chiminess is a byproduct of the fact that the tubes react quicker to the signal when it hits them. (This is often called "Slew Rate".

The Vox AC30 may be the most famous "Class A" amplifier, but some of the smallest tube amps that were geared toward beginners back in the 60's are also "Class A" and have been the secret to great low-wattage tone in the recording studio. Check out Fender Champs and some of the small Supro/Valco amps as well. These amps often have only one output tube, which is a benefit of running a small, less expensive amp in Class A configuration.

Class B refers to the fact that the output tubes are not conducting current when they are idle. You will often hear an amplifier's output circuit described as "push/pull". This illustrates how the output tubes share the job of conducting current. A push/pull circuit might have one (or more) tube amplifying the positive peaks of the signal, while the other will amplify the positive going peaks. In that case, you have tubes cycling back and forth from conducting, to resting, conducting, resting all the time. They're technically working half as hard to do twice as much work.

This gives "crossover distortion" which is what happens when the waveform switches from positive to negative. It makes the amp have a more round sounding tone, with capabilities of a more saturated sounding distortion when it's really pushed.

Class AB is an amp which operates as Class A for part of the time, and B for another part of the time. On paper, this gives us the best of both worlds. Sonically, it is really it's own thing. You would find this kind of circuit in most tube guitar amps, like the Marshall JTM 45, or the Fender Twin. It's a much more efficient circuit, and cheaper to manufacture (Class A amps need a really robust power supply to keep them up and running).

The tone is often much louder than a Class A amp, can have more clean headroom before they start to distort, but once they do, the type of distortion gets more saturated as the amp is pushed. The classic high-gain Marshall tone owes much of it's sound to the Class A/B design.

The lesser-seen "Class D" amplifier is often found in small and even battery powered solid state amplifier circuits. It is a very efficient circuit, though it has never caught on as a means of popular amplifier design. While there are plenty of great sounding solid state amps out there, the majority of tone freaks are too interested in tube tone to give much love to these amps.

Please check out some of these links for more in-depth information.

Duncan's Amp Pages (no relation to Volcanoboy)
Wikipedia page-Electronic Amplifiers
Sweetwater Sound's explanation