What is class A?

Technically:
Class A amplifiers amplify over the whole of the input cycle such that the output signal is an exact scaled-up replica of the input with no clipping. Class A amplifiers are the usual means of implementing small-signal amplifiers. They are not very efficient — a theoretical maximum of 25% is obtainable, but for small signals, this waste of power is still extremely small, and can be easily tolerated. Only when we need to create output powers with appreciable levels of voltage and current does Class A become problematic. In a Class A circuit, the amplifying element is biased such that the device is always conducting to some extent, and is operated over the most linear portion of its characteristic curve (known as its transfer characteristic or transconductance curve). Because the device is always conducting, even if there is no input at all, power is wasted. This is the reason for its inefficiency.

Simplicity:
Class A means that the amplifier amplifies the entire waveform as a waveform. There are other classes of amplifier which split the waveform into top and bottom and amplify those sections separately. There are advantages in efficiency from splitting the waveform but also a distortion that is created around the point of separation.

These are design criteria balance elements that are faced by circuit designers. Class A amplifiers are often more expensive to build but draw a lot of current, create a lot of heat and are overall not quite as efficient as some of their other letter amplifier type cousins. With more heat and power, it sometime translate to more color. Notice most class A unit has its own power supply, that just to get enough power into the unit plus keep it from too much heat.
A plain explanation, Class A circuit tend to produce even order harmonic in the tone...you know what that mean right? Cool!