NVIDIA vs. ATI
These two companies are the juggernauts of the GPU industry. Unless something revolutionary has happened between the writing of this article and the time you are reading it, those are the only types of cards you will be looking at. As I mentioned, the actual company who assembles the card may be someone else (BFG Tech, EVGA, or someone else), but it will be clear if it is an NVIDIA or ATI card.
On pure speed and power, the current champ changes back and forth depending on which company released their new card most recently. For the most part, they are pretty equal when it comes to power and performance.
What is better about NVIDIA?
NVIDIA has a reputation of better driver support and less conflicts with games and software. They are also the leader in market share, which means gaming and software companies tend to be a little more certain their products work with NVIDIA cards.
What is better about ATI?
ATI cards tend to be a bit cheaper, which is especially useful if you plan to use two cards (see the SLI and Crossfire section below). While they are a little slower to get driver updates out, and conflicts are more common, problems are infrequent. When there is a conflict, usually the software maker or ATI resolves it quickly.