The additional memory allows for more room for textures and other data that the graphics card uses. If the GPU needs more memory than it has, it will pull data from the system memory, which is slower, leading to degraded performance.
At higher resolutions, and in some games with high resolution texture packs, there are some cards that will benefit from having more than 1GB of memory. For current generation cards that would be the Radeon HD 7850 or higher, or the GeForce GTX 660 or higher. Lower end cards will also sometimes come with 2GB of memory, but that is pretty much a marketing gimmick. Lower end cards don't have GPUs powerful enough to actually benefit from more than 1GB of memory, but lower end cards might be given more memory simply because it looks impressive on the spec sheet. Also back in the day a quick and dirty way to determine how powerful a graphics card was involved looking at the memory size. That doesn't apply anymore, but manufacturers will still saddle low end cards with lots of memory in hopes that they can con somebody into buying them.
If you are looking at buying a GPU then realistically if you expect to be playing games on it then you would do well to get a card with 2GB of Ram on it. Modern games are starting to hit memory related restrictions, while most games would be ok with 1GB still its really worth paying a little more and protecting yourself from any issues that may result from having only 1GB.
I speak from experience having purchased a 512mb card quite a few years ago now instead of a 1GB card.