How does a turbocharger work?

If you know how a jet engine works, you're halfway to understanding a car's turbocharger. A jet engine sucks in cold air at the front, squeezes it into a chamber where it burns with fuel, and then blasts hot air out of the back. As the hot air leaves, it roars past a turbine (a bit like a very compact metal windmill) that drives the compressor (air pump) at the front of the engine. This is the bit that pushes the air into the engine to make the fuel burn properly. The turbocharger on a car applies a very similar principle to a piston engine. It uses the exhaust gas to drive a turbine. This spins an air compressor that pushes extra air (and oxygen) into the cylinders, allowing them to burn more fuel each second. That's why a turbocharged car can produce more power (which is another way of saying "more energy per second"). A supercharger (or "mechanically driven supercharger" to give it its full name) is very similar to a turbocharger, but instead of being driven by exhaust gases using a turbine, it's powered from the car's spinning crankshaft. That's usually a disadvantage: where a turbocharger is powered by waste energy in the exhaust, a supercharger actually steals energy from the car's own power source (the crankshaft), which is generally unhelpful.
How does turbocharging work in practice? A turbocharger is effectively two little air fans (also called impellers or gas pumps) sitting on the same metal shaft so that both spin around together. One of these fans, called the turbine, sits in the exhaust stream from the cylinders. As the cylinders blow hot gas past the fan blades, they rotate and the shaft they're connected to (technically called the center hub rotating assembly or CHRA) rotates as well. The second fan is called the compressor and, since it's sitting on the same shaft as the turbine, it spins too. It's mounted inside the car's air intake so, as it spins, it draws air into the car and forces it into the cylinders.
Now there's a slight problem here. If you compress a gas, you make it hotter (that's why a bicycle pump warms up when you start inflating your tires). Hotter air is less dense (that's why warm air rises over radiators) and less effective at helping fuel to burn, so it would be much better if the air coming from the compressor were cooled before it entered the cylinders. To cool it down, the output from the compressor passes over a heat exchanger that removes the extra heat and channels it elsewhere.