Overclocking = running your graphics card's GPU at higher speeds than it is rated at, i.e., an HD 5870 runs at 850 MHz stock, overclocking it would be running it any speed >850 MHz.
Overclocking normally increases performance, and only has a really noticeably performance boost with a overclock of greater than 50 MHz, generally speaking.
It may damage your hardware if the temperatures are too high or you overvolt the GPU too much.
You can use ATi Overdrive for a limited overclock, after that probably the AMD GPU Clock Tool.
I've been hearing a lot about voltage during an overclock... I basically understand stuff such as the GPU temperature, the core clock (mHz), but I don't get what the voltage is... ASUS actually has voltage tweak cards that they claim to increase performance by 50%... but what's the real deal about voltage? And what's the difference between a factory overclocked card, a reference card, a card you overclock yourself, and a factory overclocked card that you overclock further (in terms of durability and value for money)? Sorry about the large number fo questions but I'm new to this and really confused... thanks a lot for the help!