A pilot light is a small flame of burning gas, often in a gas burner, or boiler, which acts as the trigger for the much larger flame of the main boiler.
The pilot light is normally on, and therefore consumes a small amount of fuel whether the main boiler is on or off. Whenever the control system requires the main burner to be on (at a specific time, or when the temperature falls below a pre-set limit, for example), the motorised valves open, releasing a large flow of gas (or sometimes another fuel). The flow is designed to pass across or around the pilot light, which ignites the main flow of fuel.
If the boiler is not required for some time, then the pilot light can be extinguished, although the procedure for re-igniting the pilot light can sometimes be awkward and time-consuming.
The alternative to a pilot light is a piezo crystal, which causes a spark to ignite the gas. You can sometimes hear a rapid tick-tick-tick noise just before a boiler comes on. That is a piezo crystal trying to ignite the fuel.
The benefit of the pilot light is that it is quiet, usually safe and fairly reliable. The disadvantage is that it might blow out in a high wind, and constantly consumes a small amount of fuel. With a piezo crystal, the noise can be a problem in a domestic setting, but the fuel consumption is marginally lower. Also, older piezo crystals would not last more than a year or two.
Most modern domestic systems are moving across to piezo systems, on the grounds that the pilot light wastes fuel, and naked flames can be potentially dangerous. Simultaneously, piezo technology is improving in terms of reliability and lifetime. Industrial systems still tend to use pilot lights, although that may also change in the coming years.
Neither system will allow the full flow of gas to continue for long if, for some reason, it does not ignite. Sensors detect the temperature of the area above the main burners, and if the flow is on, but the temperature is low, then the control gear will automatically shut off the main gas supply.
Also the power on indicator
As an extension of the above, in many electronic gizmos, computers, VCRs, etc, the little red (or green) LED which indicates that power is being supplied to the system is also known as the pilot light. (Thanks, enth)