Technically, distortion is any change in the signal as it propagates through the circuit. There is always some distortion, even in a clean tone, because there is no perfectly linear amplifier.

Real over drive is a condition where an amplifier is pushed beyond it's design limit. A pre-amp or power amp is designed to work in a linear range (not perfectly linear) for certain input signal voltage levels. When the input signal is beyond the expected level then the output signal will be distorted due to the circuitry being over driven. In just about any other field of audio this would be a bad thing. We guitarist think it sounds cool. :)

Unlike other audio applications, guitar amps are now designed to distort things on purpose. Much of what we call distortion is really an attempt to emulate a dramatically over driven signal.

Now we have devices (pedals or circuitry built into the amp) that attempt to emulate various stages of over drive and other affects. You can tell that it's not real over drive because the fuzzy tone doesn't change with volume.

Cheers,