A turbocharger is (usually) a way of getting more power out of an engine without increasing the displacement of the engine. This might allow a 1.5l engine to behave more like a 2.0l engine when extra power is required. You can think of it as a sort of temporary engine enlarger.
As Clifford said, it's basically a pump that uses kinetic energy from the hot exhaust gas when the engine is working hard to increase the pressure of the air going into the engine. Because the air is above atmospheric pressure, it contains more oxygen than the same volume of air at atmospheric pressure. This means that more fuel can be mixed with the air to produce more power, and that results in even more hot exhaust gas to drive the turbine.