An afterburner is a piece of equipment used on many modern jets which burns additional fuel in the exhaust trail to provide extra propulsion. Afterburners are most commonly found on military airplanes, though they have been used on a handful of supersonic civilian aircraft.
Jet engines operate by igniting fuel and spitting out mass behind them to create forward momentum. Jet turbines pull air in from the outside environment, compress it to a high density, add a fuel mix to the oxygen, and ignite it. This process is very efficient, but far from perfect.
When the mix comes out the exhaust end of the jet engine, there is still a substantial quantity of oxygen remaining in it. An afterburner is used to take advantage of that remaining oxygen, by dumping more fuel into it and igniting the mix.
Afterburners are extremely inefficient, and are used primarily because they are such a simple component to add to a jet engine. Rather than adding weight and complexity to the engine, one adds a relatively small injector and tube behind the engine proper. With this small apparatus added on, a jet engine can receive a boost of up to 50% on an already very efficient engine.
Since afterburners are so inefficient, they are used sparingly on military aircraft. The primary use of an afterburner is during takeoff from a very limited runway space (such as an aircraft carrier or jungle runway) and when an extra boost will give a distinct combat advantage in an air skirmish.
Afterburners also produce a large, Bunsen-burner-like flame out of the back of the jet engine, giving them enormous visual appeal. When most people think of a fighter jet taking off, the most vivid image that appears is that of the enormous torch propelling the jet along.
The term afterburner may also be used when discussing incinerators. In this context, an afterburner is a very hot flame added to the process to remove virtually all smell and remaining particulate matter (in the form of smoke) from the output of incineration.
In a more general sense, an afterburner can be viewed as any second-tier flame system, either used to utilize some bit of energy or to eliminate particulate matter from waste gasses.