The mechanical beauty of your clutch starts with a thing called the flywheel.
If your vehicle has a manual gearbox, towards the back of your engine you'll find a big metal disc known as flywheel. Bolted to the flywheel is the two principal parts of your clutch - the clutch plate which connects to the gearbox, and the pressure plate which connects to the engine.
When the clutch pedal is released, the pressure plate moves the clutch plate into the flywheel until the two lock as soon as their speeds match. It's a bit like the reverse of a brake... when you press a brake pedal you are creating the friction required, when you press a clutch you release friction.