So another variation of an effect that is similar to overdrive or distortion, but definitely a lot different is fuzz. Artist that comes to mind that uses fuzz is Jimi Hendrix, pioneered it. There was an effect called the fuzz face that he used. There's a lot of different pedals. The Fuzz Factory is a modern take on the Fuzz Face with a lot of wild and crazy variations by Zachary Vex, by Z.Vex.
Jack White is an artist that uses fuzz a lot. What the difference is in a nutshell, is fuzz really alters the wave form, almost into a square wave. Without getting too technical, it really sounds like your amp's broken. I mean I guess that's the only way to really describe it. There's a lot of different colors in between that statement. You can get it to sound almost like a very saturated, compressed distortion, all the way to something that almost sounds like an octave.
That it's so broken up and the wave form is so altered that it almost sounds like two notes being played at the same time. There's a lot of variations, but fuzz is a very distinctive tone. You'll know it when you hear it. I guess the long story short of that, is that if you are looking to get a Jimi Hendrix tone and you buy an overdrive pedal, you're probably not going to get that tone.
You're probably going to be better off getting a distortion pedal and a fuzz pedal or just a fuzz pedal and crank your amp. Just try and get the tone that you like. But fuzz, very different, probably on the far end of the spectrum, as far as how affected your tone is. I'd say fuzz is the most affected, least natural sounding. Listen to Jack White, listen to Jimi Hendrix if you really want to hear what that sounds like.