The manufacturing process used to create the chip that powers this board. Expressed in terms of nanometers. In general, the lower this number is, the newer the process is. Smaller processes can fit more transistors in less space, and generally produce less heat and consume less power all while providing higher overclocking headroom.
This number refers only to the main GPU on a video card. Other chips on the board (including the memory) will have completely different characteristics.
Does It Matter?
No, not really. Most you can really tell from this number is how new a chip is and perhaps how much overclocking headroom you can expect out of it (given that a certain process tends to top out a certain speed).
Since video card manufacturers are "fab-less" companies (i.e. they don't own their own fabrication plants), they must choose between various other fabrication companies to make their chips for them, and then choose different fabrication techniques from these companies. The size of the process is not the only option, often fabrication companies will offer different types of processes at the same size.
Different processes are at different stages of development at different times. Always going for the newest process isn't always the best idea (as evidenced by NV30 and R520) since it can lead to delays if things don't go right with the process. However, the newest process almost guarantees the highest clock speeds, so it's a bit of a gamble on the part of the chip designers as to which process they choose.
For example, both the GeForce 6800 Ultra and the Radeon X800 XT were built on a 130nm process. However, the X800 used TSMC's somewhat experimental "low-k" process, whereas the GeForce used IBM's regular 130nm process. The result was, the X800s overclocked much higher, but had a much higher failure rate. So while nVidia's chips were limited to about 400Mhz, they had a much easier time producing chips with all 16 pipelines working.