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.
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.