Texture Memory
Texture memory is the portion of the video memory used to store textures. If there isn't enough texture memory to store all the textures needed in a scene, the video card will have to retrieve the textures from system memory. This is a very slow process, almost crippling a player if this happens during a deathmatch.
Dedicated frame buffer
Older processors like the 3dfx's Voodoo/Voodoo2 and Intel's i740 have split memory architectures where frame buffer memory and texture memory are separated. This prevents the texture memory from claiming unused portions of the dedicated frame buffer.
Unified Memory buffer
Current cards now have unified memory buffers that allow the frame buffer and texture memory to share the video memory. The frame buffer gets first dibs, and texture memory gets the leftovers. The larger the frame buffer, the smaller the texture memory. Higher resolutions require large frame buffers, reducing the amount of texture memory available.
Double, Triple, and Quad buffering
Double buffering is the process of using two frame buffers for smooth animation and video playback. While image of the first buffer is being displayed, the video card renders the next image on the second buffer. Once the second buffer finishes, the buffers are flipped, and the process repeats. Triple and quad buffering is just like double buffering but with more buffers. Animation and video playback appear smooth because only complete images are displayed. Multi-buffering increases memory needs; double buffering doubles the size of the frame buffer because two frames are being used.
Size and speed
So you can imagine that a large portion of the memory on a video card is dedicated to texturing, and the speed of this memory (and the on-board data bus) is of utmost importance to most video cards. Faster and larger video RAM can make all the difference in a highly-textured environment, especially in high resolutions. Add to this large amounts of addition (semi-fast) memory available through AGP texture streaming, and you can start to see what the big deal is about.
However, we're not quite done with memory yet!