What is distortion?
Distortion is kind of the harder, more evil cousin to overdrive. The technology is pretty much the same but the distortion is clipping the peaks and valleys of the soundwaves much harder than overdrive, which also generates more harmonics than an overdrive effect would do. Distortion pedals simulates what happens when you drive full-stack tube amps really, really hard.
Any good examples?
Very much modern rock, hard rock and heavy metal is built around distortion, but it is also used in pop music. Johnny Greenwood of Radiohead is not shy to plug in his Marshall Shredmaster and the Pro Co Turbo Rat can be heard on songs by both Blur and Suede. But to really hear clear distortion, listen to a song by Metallica, Pantera, Nirvana (at least in the chorus of their songs), My Chemical Romance or just about any other rock or heavy metal band that you like from the eighties and forward.
Which pedal should I get?
This depends a lot on what music you want to play. A rock guitarist would maybe not use the same pedal as a death metal guitarist. At least not with the same settings. I've already mentioned the Pro Co Rat and the Marshall Shredmaster but there's a lot more good distortion pedals: the Boss pedals DS-1, DS-2, HM-2, ML-2 and MT-2 (the three latter being the heavy metal choices), the MXR Super Badass Distortion, the Ibanez Jemini Distortion and the DigiTech TL-2 are some of them. Remember that many guitarists uses the distortion from their amp, but there are probably just as many that settles on one (or several) of these pedals to create distortion.