QUICK ANSWER: A thermocouple lets the pilot light in your water heater stay on. When the thermocouple is operating properly it will tell the thermostat to turn on the main burner. The main burner is what heats the water in your water heater. The thermocouple is a very simple part but it has a very important job. If your thermocouple is not operating properly your water heater will not turn on.
TECHNICAL DETAILS: A thermocouple is a temperature measuring unit that conducts energy between two dissimilar conductors. The unit is usually made of copper and a different metal. When one end of the thermocouple senses a different temperature it will create a small amount of electricity. I.e. one end in the pilot flame and one end in the thermostat. The millivolts are registered by the controller and allow the main burner to ignite. When the thermocouple is malfunctioning it will not keep the pilot light lit. When the pilot light is not lit, the thermostat will not allow the main burner to ignite.