Audio Compression is the process of lessening the dynamic range between the loudest and quietest parts of an audio signal. This is done by boosting the quieter signals and attenuating the louder signals.
That's what most people think it is, but it's more complex than that. For the casual user, compression is considered a volume boost, for the novice, it's often a nightmare, and for a very expressive player/singer ... it could go either way. Most singers I record don't use any.