Generally speaking a 256-bit option is better. It refers to the width of the memory bus on the card, so a higher memory bus width means more data can be transferred between the memory and the graphics processor, running at the same speed as the lower bus width. So a 256-bit memory bus can deliver twice as much graphics data to the processor as the 128-bit at the same speed.

This is something you should watch out for when you are thinking about buying a low-cost / budget card. Sometimes you will have to figure out whether you want more GDDR memory but with a lower memory bus width vs. less GDDR memory with a higher bus width. The first option gives you more memory, but as your games/applications pull data from the memory, it will be delivered slower over a lower bud width. The second option will give you less overall memory, but the bus width is higher, so the performance might be improved in that way.

Do a bit of extra reading around if you are thinking about buying a graphics card, and of course, ask as many questions as you like :-)