If you're a rocker you'll jump to guitars, a drum and bass head then you'll spring to some violent synths... a studio engineer and you'll head for the nearest noose. It's certainly a versatile technique. But where did it come from?
Conception
Arguably the first time distortion was used creatively on a recorded song is a 1950s blues song snappily titled "Rocket 88" - where apparently a broken amp resulted in a distorted guitar tone. So next time your gear breaks, don't get frustrated; chances are you've just been handed the next big sound!
Sound engineers such as Joe Meek pioneered the use of ruining their audio with distortion for effect. Home-made electronics was the way to go back then, since they couldn't exactly pop down to their local music shop and get a cheap distortion pedal. Conventional studio engineers would have called people like Meek and Phil Spector (another producer who conceived the "Wall of Sound" idea of multi-layering instruments) mental at the time. Maybe they'd've said their minds were distorted more than their guitars?
Though I guess they would have been right about them both. Meek ended up killing his landlady, then himself and Spector kinda killed a prostitute recently. Maybe she made fun of his hairdo (which, incidentally, looks like he spent a bit too much time playing around with distortion.)
Back on track, bands such as Cream and the Jimi Hendrix experienced popularised distortion for use with guitars for a cool tone, proceeding to influence acts such as Black Sabbath who in turn popularised it for use with heavier music.
Companies such as Fender and later on Marshall made this possible by bringing out their ranges of purposefully overdriven amps to achieve that magic new distorted tone.

Types of Distortion
There are quite a few types of distortion about - let's take a look at some of the most famous / infamous.
Tube Distortion
When someone pushes the power tube of their tube amp too hard - This is generally for the more overdriven, warm sound. It is emulated by pedals by pushing too much volume into their transistors. Tube amps are widely regarded as the most desirable by guitarists for the quality of the tone.
Preamp distortion
When you push too much volume into an analog preamp - this can really shape the tone of an effect, and used in unison with other distortion types, it can significantly alter the sound.
Feedback
When an output is put into the input of the same device, a high pitched sound is created called "Feedback" - used extensively by obnoxious rock and metal guitarists for years, it can be used to great effect.
Digital Distortion
This is widely regarded as a nuisance, but has recently come into musical maturity with genres like glitch and noise music (What an oxymoron) integrating themselves into other genres. This is achieved when either you record too loud of an input into your audio sequencer and the audio file "clips" where it can't store the volume of the audio, causing a distorted effect, or it can be achieved via circuit bending - where electronic devices (from guitar pedals to children's toys) are short circuited to see what noises they make.

Where is it now?
Distortion has evolved into a fine art. Companies such as Line6 and Native Instruments are pushing the boundaries with Guitar Amp Simulation software and hardware. This gives people the power to change their distortions on the fly if they don't like the initial recording - an invaluable tool to any producer.
It has also become synonymous with some of the heavier electronic music genres - Drum and Bass, The Heavier Hip Hop and Industrial music all use it to beef up their sounds.
Death Metal bands continue to push the boundaries of what is noise and what is music with their often painfully overdriven guitar tones, Electronic musicians do the same by applying so much distortion to their beats that you can't tell what was a kick and what was a hi hat any more.
Effective use of distortion has opened up creative doors for everyone wanting to add an edge to their sound - a beefiness that was lacking with clean tones. Vocals can be made more intense, samples can be given a new lease of life - there are people pushing the boundaries so much that you might thing they are mad for considering their creations "music".
But just think - in 40 years they might be the ones who are millionaire prostitute killers! Who know's!
