An afterburner is a secondary combustion system which burns additional fuel downstream of the combustion chamber, to further increase thrust at the expense of much higher fuel consumption.
This is the Pratt & Whitney F100 afterburning turbofan, variants of which power the USAF's 4th-generation fleet of F-15s and F-16s:
The final spoke-looking thing, plus all the space between the turbine core and the exhaust nozzle, is the afterburner. In this area, fuel is sprayed directly into the exhaust stream, where the heat from the air leaving the turbine core is enough to ignite it. This additional pressure adds to the thrust produced by the turbine.
As I said, though, the tradeoff is increased fuel consumption, sometimes dramatically so. The F-16 at full military power and low altitudes burns about 8000 pounds of fuel an hour, which with a full droptank configuration gives it about 2 hours' flight time. Cruising at higher altitudes, that flight time can be further extended as both the higher altitude and the lower throttle setting (about 80%) reduce fuel flow rate by up to 40% versus low-altitude flight.
In full afterburner at low altitudes, the F-16 can burn in excess of 64,000 pounds an hour. At full throttle, a U.S.-variant F-16 with maximum external fuel stores has about 20 minutes until it's on emergency reserves (which would only last an extra minute or so at full afterburner). The speed gain is minimal; the F-16 cruises at between 450-550 knots, while full afterburner only increases that to about 700-800 knots with a typical underwing loadout. So, burning 8 times the fuel, you get around a 50% speed boost.