distortion is actually something that is NOT wanted in amplifiers. But guitar players found that this default sounded good.
When you make a guitar string move, and you amplify it, you should get the same sound, right? Well you can never amplify without coloring a little. If you amplify a lot (more than what the amplifier can) the sound will still go through the same devices, that will have to leave some clips of the sound out of the speaker. That's distortion. Usually, we say that tube amps have a smoother distortion because they tend to smooth out how they cut the clips of the sound they leave out.

(ps: I have used a very schematic way of explaining it without physics or whatever, but if you have what it takes, some people here can write you a book about distortion)