OK, here's the basics:
Your motherboard has a CPU and RAM, connected to each other by a very fast connection. The video card plugs into the PCIE bus, which is a very slow connection when compared to that CPU <-> RAM connection. So the video card, which is like its own little motherboard with a GPU instead of a CPU, needs its own RAM as well - called Video RAM, or VRAM.
The VRAM connects directly to the GPU so that they can communicate at high speed. If you think of this connection like a pipe, the 256-bit number of the video card is a measurement to how large in diameter that pipe is. The larger the number, the bigger the pipe and the more information can be sent between the VRAM and GPU at once.
1GB refers to how much VRAM is physically on the card, and how much information it can hold. VRAM is used for stuff like holding the textures in a game, and these texture files can get very large and take up a lot of space when you set a game's textures to the highest settings (depending on the game). For new games, 1GB will not be enough to use Ultra or even High resolution textures in most cases. Other things liek your monitor resolution (esp gaming on multiple monitors) and some types of anti-aliasing (like MSAA) will also increase the amount of VRAM you need
If the files that need to be stored on the VRAM won't all fit on the video card, they will have to be stored on the main RAM instead. The problem is that now when the GPU needs them, it has to go over that very slow PCIE bus to get them. Each time this happens, your game will stutter.
I can run a 290/X with Elpida vRam at approx 1250 x 1500Mhz x 132 x 50 bandwidth but if I had Hynix or Samsung vRam your talking another story in regards to configuring OC'd frequency and VDDC off-set voltage for the card with proper GPU and VRM cooling.
Give me Mosel vRam anything but Elpida!
Driver prominence Vs Memory Leaks - Hum!
Old OC'r Experience - PERIOD.
Cmon man, he asks what is VRAM and you come at him with this? Now you're just trying to confuse the guy.
Also, I'd be willing to bet that the amount of people here (even of the very well informed users) that factor the brand of VRAM used on a video card into their purchase is tiny. I suspect buyers most expect that if they are getting a card from a respected manufacturer, it will come with reputable memory chips.