Noticed that our hot water wasn't hot. The pilot light was out. We re-lit the pilot and flame came on. Water heated up and then the flame went out as did the pilot light. Why doesn't the pilot light stay on?

How old is the water heater. What brand is it, what size is it. If its fairly new, it is sealed combustion. How long does the flame stay lit? It may be plugged intake filters, or faulty thermocouple. The new water heaters have an extra safey shut off device. Kind of hard to trouble shoot from here, without being able to see operation first hand. Let us know exactly what is happening, how long burner stays lit, if flame changes consistency, and any other info you can give us.