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Nvidia/Geforce, ATI/Radeon are the Developers/manufacturers.

Essentially the first choice a buyer makes is deciding whether they want to go with an Nvidia/Geforce graphics card or an ATI/Radeon card. They use different chips and technology. (Kinda like deciding whether you want a corn chip or a potato chip. They will both serve the purpose of a snack, but each has its own qualities)

The funny numbers you see tell a story of a cards quality. (e.g. the ATI 5750, 5770, 5850, 5870)
The first number tells you its series 5 (the latest, as opposed to the last series which were 4XXX so the number tells you it is modern) the other numbers simlpy tell you its quality - the higher the number the better the card.
The second number is the most significant (lets call it the 'generation')- the 5850 and the 5770 for example, the 2nd number tell you that the 5850 is considerably better than the 5770 as it is a higher generation of card.

The third number helps identify the cards quality within its generation, for example the 5770 and the 5750, the third number tells you that the 5770 is better than the 5750.

Nvidia have their own numbering system but it is a similar idea, for example their latest series is 400, the two cards are 470 and 480. You can see by the numbers that the 480is the better card. Their last series was the 200, so all their cards began with 2 and there ws a 240, 250, 260 etc. again, the second number tells you which is better.

XFX/Sapphire/Gigabyte/Diamond etc. This is the 'brand' of card. Once you have chosen which developer/manufacturer you are going with and the card series/quality you now have the different brands, often the main difference between each one is things like customer service and the warranty offered (unless they have tweaked with the manufacturers card, it is essentially the same card regardles of which brand you pick, the only differences may be custome heatsinks or slight clock speed adjustments).
(This is like deciding you are buying a blue sweater, you are now deciding whether you want it made by Ralph Lauren, or Tommy HIlfiger)

^+1000

A very detail explanation...  

Very well done! Sticky Worthy? : )

The talk of potato chips made me hungry  

thanks a lot, now i am aware much better than before.