Kurumuka gave a great explanation. Class A and AB can be both balanced and single ended. Both are push pull and can be tube or ss. There are some Class A SETs - single ended triode or single ended transistors where the +/- is handled by one tube or transistor. These are always single ended (RCA) and are almost always very low power.
Class A amps, because of the constant current, tend to be very fast and responsive. They also waste 75% of the power going through them and run very hot. Many at 130 degrees and in a small space, this is a huge issue.
It also matters little how hard your drive these with the exception of low wattage, Class A tube SETs. Because current running through the amp is constant, it doesn't make a difference if the amp is at 10 or 100, the amp is being "driven" the same.
One issue is that they tend not to sound right until the get scorching hot and this can take some time. Tube amps often have direct heated triodes which get them to temperature faster. SETs are an exception and unless you are in the market for a low wattage SET, I will not share further.
Class ABs are much more efficient, run in Class A mode to a point and then switch to class B. This can vary wildly from amp to amp. When driven hard, these can get very hot and you do not want to be pushing these to high volumes. They can also seem a little slower than a Class A because of the non-constant current. In practice, this is not a huge issue unless the amp is poorly made.
Class B tends to be highly distorted and there are very, very few pure Class B amps on the market. AB amps switch from Class A to Class B so that it can deal with increases in power needs with changes in volume.
If you have an amp that has a switch from pure class A to AB (borg dune and warp), your speakers and power needs drive what mode you should use.