What is a 'fast' amp?

Not asking about specific amps, rather what does a 'fast' amp sound like vs a 'slow' one? This terminology is thrown about a lot and I don't know how to tell the difference.

I got to thinking... is an amp fast when it doesn't rush the music (ie, it is faster than the music, so the music appears relaxed) or does it rush the music, making it sound fast? I peronally would think that neither is too good, but I would take the relaxed over the rushed sound, of course.

?? Perplexed.