An afterburner is an extension of a jet engine, used in military aircraft. It injects fuel into the heated exhaust pipe of the engine. It will react with uncombined oxygen in the exhaust, which increases the exhaust temperature and, thus, its velocity, giving the plane additional thrust. Afterburners use more fuel than the normal turbine, so they are used only when launching the airplane from the ground or an aircraft carrier, or when high speed is of the essence. The only civilian aircraft using afterburners is the Concorde. In addition, development of supercruise engines* has lessened the need for afterburner use. *As a BTW A supercruising aircraft is able to cruise at supersonic speeds without the use of afterburners. Aircraft with supercruise include: F/A-22 Raptor Eurofighter Typhoon Concorde Although supercruise is currently fashionable, it is not new; the first aircraft to exceed Mach 1 in level flight without afterburners was the P.1 prototype of the English Electric Lightning on August 4, 1954. When aircraft without supercruise travel at supersonic speeds, they use up fuel very quickly, and must be careful not to run out. This means that if a supercruise-capable fighter is attempting to intercept a non-supercruise-capable aircraft (or vice versa), the supercruise-capable one will be at an advantage.

In addition to Andersen's brilliant answer, I'll give a little bit of technical info on them. Afterburners are a series of rings of fuel nozzles placed behind the turbine of a jet engine. They operate by injecting fuel into the superheated exhaust air of the engine. The effect of afterburners can vary depending on whether the engine is a turbojet or a turbofan. Turbojets have all of their air going through the engine itself, whereas turbofans have a duct around the engine that bypasses some air. Some turbofans channel as little as 10% of the total air through the bypass. These are called "low bypass ratio" engines. Others, like what you see on passenger jets, pass as much as 70% of their total air through the bypass. These are high bypass engines. Bypass air hasn't been burned, so it still has its full concentration of oxygen. Bypass air joins up with exhaust gas before the afterburner, so the amount of oxygen in the merged exhaust gases is greater. The fuel nozzles in the afterburner inject highly atomised fuel (so finely atomised that is is a fog) into the exhaust stream. The heat ignites it and gives a huge boost to the pressure of the gas coming out the back. This causes a great increase in thrust - anywhere up to a 100% increase. The trouble is that the combustion of the afterburner fuel gives off vicious shockwaves that would actually tear the engine apart, so they put in a long cylinder called a 'screech liner', which is perforated with lots of small holes and 'feathers' in the metal to dissipate this energy. That's why afterburning engines have such a long exhaust pipe - it's the screech liner. The nozzle on an afterburning engine is usually adjustable - converging and divergine (closing and opening). If you watch the opening credits to Top Gun, you can see this in action. The nozzles on the F-14's open and close. In military power (without afterburner), the nozzles actually close as power increases - this creates a reduction in volume, which increases exhaust velocity. However, you want as little back-pressure as possible with afterburner, so once afterburner is selected, the nozzles will open out again. There are different styles of afterburner on different aircraft. Most American aircraft have afterburner with five power settings - they have five rings of fuel nozzles that can be sequentially activated. The different power levels are referred to as 'zones', from Zone 1 to Zone 5 (full a/b). So in that super-realistic movie Top Gun (sic), when Viper says "you can select Zone 5, and extend and escape", he's saying to kick full afterburner (zone 5) and get the hell out of there. Most older Russian aircraft (and the Concorde too, I believe) had only one power setting - on or off. In the UK and some other European countries afterburner is referred to as Reheat. Hope this hasn't been too confusing.