Why will my pilot light on the gas water heater not stay lit for more than 15 minutes?
It could be that the Thermocouple has failed or is failing. This would cause the system to think there is no flame present, and shut off the gas supply for safety (so it doesn't fill the house with gas).

If you don't feel comfortable working on the heater; or you don't have the knowledge and/or tools to do so, call a trained professional to come take a look.