Graphics card manufacturers make it hard to compare graphics cards just by their names. As long as you are staying inside a series (or generation) of graphics cards, usually bigger numbers mean also faster cards. But this does not hold true if you compare cards between different series or even manufacturers. Both, AMD (former ATI) and Nvidia usually change the major number in their naming scheme between generations (e.g. Radeon 5xxx to Radeon 6xxx).
The first, most important step for you is to decide on your requirements. Your display resolution is very important, the higher resolutions are more demanding on your graphics card. The most important part is what kind of games you play. Casual games can usually be played with any halfway recent graphics card. The requirements vary a lot between games, the newer the game the higher the requirements are most of the time.
The amount of RAM on the graphics card is not a good indicator for performance. You only need to pay attention to it when you're using a very high resolution display, as the used RAM depends strongly on the display resolution. If you have a 2.560 x 1.600 display you need all the RAM you can get, else you won't see a difference most of the time. The other exception are some mods for certain games that have enourmous demands for RAM.
The best way to decide which graphics card to buy is to look at a variety of benchmarks performed at the resolution you want to play at. Ideally, the benchmarks should be for the games you want to play, but if they are for games of the same age and type they should be enough to gauge the performance of the card.
Some resources for reviews are: