Technically any difference, apart from volume level, between an original signal and one that has been processed. Distortion commonly refers to range of raspy sounding intended effects such as Fuzz. A mild form of distortion is associated with Tubes producing a warmer fatter tone. There are also numerous undesirable forms of distortion such as Wolf tones.
Distortion can come from different stages within an amp. This includes contributions from the preamp and tone stage, the power amp, the Output transformer in the case of a tube amp, and the speakers. How these distortions are proportioned and balanced is part of what gives any amp its distinctive character.
Harmonics, which become more evident when an amp is overdriven, are also a form of distortion. There are good Harmonics that add to the warmth of an amp, and bad harmonics that can sound harsh: Tung-Sol.com:: FAQs Why is there a difference in Tube and Transistor sound?.