Ever seen the bellows used to get fires started? They work by introducing more air (or the oxygen in it, specifically) to the combustion process. Automotive engines have their equivalent, and it's broadly referred to by the term forced induction, which comes in the form of supercharging and turbocharging.
Put simply, each of these is a compressor that forces more air into the combustion chamber, which in turn means proportionally more fuel can be injected into the mix, making for a much bigger bang – and therefore more power.
The basic difference between a supercharger and turbocharger is that the former is – in most cases – belt-driven by the engine (much like your car's alternator or air conditioning), whereas the latter features a turbine propelled by the motor's exhaust gases.
Turbocharging is nowadays far more prevalent because it's the more efficient solution – s it doesn't require engine power to spin it. That said, there is a bit of "lag" associated with turbos, because they only deliver meaningful boost at mid to high engine revs.
Some manufacturers get around this by resorting to twincharging, which combines a supercharger and turbocharger – the former to generate boost at low revs, the latter at higher engine speeds.