A class A amplifier is, simply put, one amp pushing a speaker in an out. The best way to describe it is to relate it to Class A/B. Class A/B amplifiers have 2 amps, one to push the speaker out (+), the other to pull it in (-). In guitar amps that use tubes, this means that one tube, a driver tube inverts the signal polarity before it reaches the B side.
A class A amp pushes and pulls the speaker, and thus must do more work. Most commercial (in home) amps (stereo receivers, etc.) are class A amps. Class A/B is generally used in guitar amps over 30 watts, high powered Bass amps, and professional power amps. Class A/B generally delivers the same power at double the impedance. So if an amp gets 100w per side @ 4 ohms, then it will generally get 200w bridged mono @ 8 ohms. This is rarely 100% true in practice.
Class A/B is also called "Bridging" an amp (transistor amps generally).