Short Version
The number of transistors in the chip on the card.
Long Version
The number of transistors on a chip isn't really relevant in terms of performance. However, it does give an indication as to the relative complexity of the chip. This can give an indication as to how much power the chip will need, and thus how much heat it will produce.
The number of transistors used in the chip is displayed mostly for curiosity sake. You shouldn't consider the number of transistors an important factor when comparing cards.
