Short Version
The speed at which the card can access memory.
Long Version
Memory bandwidth is equal to the size of the memory bus multiplied by the speed at which the memory is clocked.
The higher the memory bandwidth, the better the card will be able to handle large textures and anti-aliasing and anisotropic filtering. Not to say that you don't need a lot of memory bandwidth if you don't want to use these features, you still do. Memory bandwidth is important in nearly every part of graphics processing.
The amount and speed of the memory matter very little in comparison to the overall memory bandwidth. If you want a card with good memory, this number says it all.