Thermocouple (or Thermopile) is bad.
If you look at the pilot flame, it is heating a little brass looking stick with a couple of wires coming out of it. The is the thermocouple that tells the gas valve to keep the pilot on. It is simple to replace and costs about $15. Your local hardware store should have it. The thermocouple is skinny like a toothpick, if it is fatter, like a pencil, you have a thermopile which costs about $30-$40. Just take what you have into your hardware store and they will set you up. Besides disconnecting the wires (remember where they go!) you have to either unscrew it out the bottom or just pull in down if not screwed in.
Good luck!
Rick
November 2008