The clutch is the part that basically connects the engine to the gearbox.
Basically, from the left is where the engine would be, and the right is where the shaft connects to the wheels.
The clutch, when engaged, disconnects the engine to the gearbox so the gears (gear ratio) can be changed without damaging the gears.
The clutch is operated using the most left of the 3 pedals of a manual car. If the car is automatic, it’s slightly different.
For what exactly is inside the clutch, there are 3 main parts.
The flywheel connects to the engine.
The friction disc connects to the transmission, and touches the pressure plate and flywheel. It’s material is similar to the material used in brake pads.
And the pressure plate, which is connected through the friction disk to the flywheel.
When the clutch is engaged (foot off the pedal), the springs in the clutch press the 3 parts together, meaning that the engine and transmission is connected, spinning at the same speed.
When the clutch is disengaged, a piston pushes the release fork (far right) which in turn pushes the clutch release bearing, which pushes the middle of the diaphragm spring.
As the middle of the diaphragm spring is pushed in, the edges push out, and pins on the outside of the disc pulls the pressure plate away from the clutch disc. The clutch is now disengaged, meaning the flywheel and the pressure plate spin differently from the friction disk. No torque is transmitted from the engine to the gearbox.
(now there is a small gap between the pressure plate and the clutch disk, compared to the earlier image)
There are a lot more different types of clutches, like the electromagnetic clutch, but this is probably the most common.