The memory quoted for each card refers to the graphics card's onboard RAM capacity.
This RAM is usually DDR3 and is generally a good deal faster than the pcs own RAM.
The graphics cards make use of this fast RAM to load information they need to access frequently/quickly and this helps give the best performance and smooth gameplay.
As to how much memory you need......well that's a bit more complicated.More onboard RAM does not necessarily mean a faster graphics card.A lot depends on how powerful the graphics card is.With a high end card you can use the highest detail settings/resolutions/effects and that needs more information loaded into memory.......256MB is a minimum here (the very latest top spec cards are now starting to appear with 512MB)
Now a 6600GT is a good, fast graphics card,but it is nowhere near as powerful as the high end cards.This means that the settings it will run at are lower and as such it will not necessarily benefit from a lot of onboard RAM.The chances are that there will be no gain in performance from using 256MB over 128MB with a 6600GT.
If,on the other hand you can find a 6600GT with 256MB RAM that is close in price to one with 128MB,buy it.
If there is a large difference in price then it might be better to go for the cheaper 128MB card and spend the difference increasing your pcs RAM.
One thing that is becoming very noticable is that a lot of the latest games need a minimum of 1GB system RAM and preferrably more.