The classification of "A" is a technical classification.
Class A amplification, be it mic pres, line pres, or power amplifiers means that the top half and the bottom half of a single cycle is reproduced by the same output device continuiously.
In most electronics, you have transistors (or tubes) that hand off 1/2 of the wave to a simalar device to do the other half of the wave. In a "Push Pull" amplifier, one set of transistors (PNP) does the top half of the wave, then the other set (NPN) does the bottom half of the wave. This is called the switching Output circuit of amplification. Take a frequency of lets say 10,000 Hz. or 10,000 cycles per second. (10KHz.)
In a push pull or non class A design, this is switching 40 thousand times to produce the 10K note in 1 second. (Switch on PNP, Switch off PNP, Switch on NPN, Switch off NPN) This switching causes what is called "zero crossing notch distortion" as the switching causes a small (most of the time inaudible) "blip" in the waveform at zero crossing.
Class A devices do not switch at all and produce the entire cycle, giving a smoother flow to the waveform. The result should be a more accurate reproduction of the wave and smoother, more coherent sound.
Since class A output devices do not receive the 50% rest state as non class A circuits, they run much hotter as far as temperature. Also the raw components designed to be used this way are more expensive and thermodynamic coupling in the physics of the design must be considered.
Hope this helps.
Do a web search on different classes of amplification. You have Class A, AB, B, D, G, H, H+...many different topograpic choices avalible to designers. Class A/B and class B units' can be designed to sound great given the proper capacitor coupling and filtering.
Beware of the cheesy companies that could possably (although I do not know of any) use the term class A for something else.
A popular audiophile magazine classifyes equipment quality in classes of A,B,C..etc.. that has absolutly no bearing on the circuit design, causing confusion amongst other things, to some.