There are only two (2) manufacturers of GPUs that we care about. Nvidia, and AMD. They make the actual GPU (graphic processing unit), the little silicon die that does all of our graphic work.
Nvidia makes GeForce, AMD makes Radeon. That's just the name of their desktop lines. Nvidia also makes Quadro and AMD makes FirePro, but those are for business use, and we don't care about those as much for home user.
Everyone else buys these chips from Nvidia and AMD, and build a full card out of it (I'm 90% sure no one else actually manufacturers the chips on their own contract). So you cannot buy a Nvidia card or a AMD card. You buy an Asus, EVGA, MSI, etc, which has either a Nvidia chip or an AMD chip on it.
When you start buying cards, you look at the chips themselves. Right now, for example, the two most powerful chips you can get are based on a Radeon HD7970, or Geforce GTX 680. The next two, down the line, are Radeon HD 7950 and Geforce GTX 670. Then you take the actual chip, and you look at which manufacturers you want to buy.
Say you want the 680. All 680-based cards are roughly the same in performance/power/clocks at the beginning. However, certain cards will be clocked faster (meaning they do more work) from the factory. However, in an ideal world, you can take any 680 card and overclock it to the same power. The difference is that certain manufactuers will build better cards around these chips, meaning they can pump more power into the chip, cool it better, which means in the end, it's a faster card overall. You will pay for that extra overclocking ability. This is why an Asus 680 or a MSI 680 costs $100 more than a basic Galaxy or Zotac 680. They build their own boards from scratch (instead of using Nvidia's recommended design), and use better parts and coolers. Whether that is worth an extra $100 is debatable.
If you look at reviews, they generally review the stock version of the chip, as designed by nvidia or amd. This will be the slowest version but it is guaranteed to always run at that speed. This way you can compare the baseline. Any overclock is considered as a bonus.