A turbocharger is an exhaust-driven turbine that boosts an engine's power output.
Normally, the downward motion of the pistons pulls air into the engine; this air is mixed with fuel, which is ignited to make power. Stepping on the accelerator increases the amount of air that can be drawn in. (So you're not really stepping on the gas; you're stepping on the air!)
A turbocharger uses a pair of turbines (a type of fan blade) on a common shaft. One turbine is piped to the exhaust, the other is piped to the intake. The flow of exhaust spins the exhaust turbine, which also spins the intake turbine. The intake turbine blows air into the engine at a greater rate than it can pull in. The greater volume of air can be mixed with a greater volume of fuel, increasing power output.
In order for the turbocharger to work, there needs to be enough exhaust pressure to spin ("spool up") the turbines, so they don't really start to boost power until the engine is spinning at 2000-3000 revolutions per minute (RPM). This is called turbo lag. Once the turbo spools up, look out -- the result is usually a strong surge of power, accompmanied by a jet-engine-like whistle.
Turbos are a lot of fun and well suited to sporty cars, but their wind-up-and-go power delivery can make for a neck-snapping ride. That's why family and luxury cars generally use larger engines to develop more power, instead of a turbocharger.
