The basic idea is that the exhaust drives the turbine (the red fan), which is directly connected to (and powers) the compressor (the blue fan), which rams air into the engine. For simplicity, we're showing only one cylinder. Here then, in summary, is how the whole thing works:

Simplified diagram showing the component parts of a turbocharger and how they work

1. Cool air enters the engine's air intake and heads toward the turbo compressor housing.

2. The compressor fan helps to suck air in.

3. The compressor squeezes and heats up the incoming air and blows it out again.

4. Hot, compressed air from the compressor passes through the heat exchanger, which cools it down.

5. Cooled, compressed air enters the cylinder's air intake. The extra oxygen helps to burn fuel in the cylinder at a faster rate.

6. Since the cylinder burns more fuel, it produces energy more quickly and can send more power to the wheels via the piston, shafts, and gears.

7. Waste gas from the cylinder exits through the exhaust outlet.

8. The hot exhaust gases blowing past the turbine fan make it rotate at high speed.

9. The spinning turbine is mounted on the same shaft as the compressor (shown here as a pale orange line). So, as the turbine spins, the compressor spins too.

10. The exhaust gas leaves the car, wasting less energy than it would otherwise.

In practice, the components could be connected something like this. The turbine (red, right) takes in exhaust air through its intake, driving the compressor (blue, left) that takes in clean outside air and pumps it into the engine. This particular design features an electric cooling system (green) in between the turbine and compressor.

Simplified diagram showing the component parts of a turbocharger and how they work

Artwork: How the turbine and compressor are connected in an electrically cooled turbocharger. From US Patent #7,946,118: Cooling an electrically controlled turbocharger by Will Hippen et al, Ecomotors International, granted May 24, 2011. Artwork courtesy of US Patent and Trademark Office.