The memory bus is the channel of communication between the processing unit and the main memory.
In the case of video cards this is between the GPU (Graphics Processing Unit, which does all the graphics calculations) and the Graphics Card memory (which stores data to do with the graphics, such as textures and so forth).
The memory bus has two main attributes: Width, and speed. Think of it as a road, with cars travelling along it. The faster the bus is, the faster the cars can drive.Thus the more cars can drive along it in a certain time. Translated this means more data can travel across the bus (improving performance) within a certain time frame. Also, how fast a road is depends on its width. The wider the road, the more cars can travel along it at a given speed. Same with the memory bus: the wider it is, the more data can travel along it at a time (improving performance).
Thus the faster, and wider (ie more bits) the memory bus is, the faster and therefore better it is. A 256bit bus is better than a 128bit bus, all other factors being constant.