The nanometer, shortened to nm, is a measure of how large the transistors are fabricated. By shrinking the fabrication process more transistors can fit in the same area, so CPUs can become more powerful (in a somewhat vague sense).
The problem in shrinking down the fabrication process is two-fold.
First, materials need to change. Think of the electron as a bullet, constantly being fired at a gate; this image is meant to show what a transistor is, a gateway to either allow eletrons to pass or to stop them. A seven meter thick wooden wall will stop the bullet, but as you have less wood the bullet comes closer and closer to penetrating. By shrinking down fabrication, or in the analogy decreasing the wood, you will need a stronger gate material, as well as a weaker bullet (think smaller voltages). Traditionally doped silicon worked for older processes, but Intel and AMD both now use high-k substrates. This technology can allow current manufacturing techniques, but how much longer until it doesn't work is a topic for vigorous debate.
Finally, you don't build a skyscraper the same way you build a garage. The ability for a given architecture to scale as it gets smaller is a real question. If you read up on Bulldozer, you can find out about an experimental new architecture and smaller manufacturing process all in one chip. The problem there is that new architectures often require several revisits before they are successful and mature. New and unique architectures will be needed in the future, so that all those extra transistors can be utilized.
The net result is that smaller processes produce better quality chips in general. There are some exceptions (this may incite flaming, but I think that AMD's Phenom is still superior to Bulldozer), but that rule is generally one to live by.
Note:
Intel has a tick-tock release cycle. They introduce a new architecture, then shrink the die. Subsequently, the next chip will have the same process size as the shrink, but be a new architecture. For reference:
Edit:
No credits, my bad. Check out the following for the original source:
http://en.wikipedia.org/wiki/Nehalem_%28microarchitecture%29