The clutch assembly consists of the pressure plate (left) and rotor. These steel components clamp together when the clutch is engaged to send pulley drive to the compressor unit. When the clutch is disengaged, the pressure plate springs forward, and the rotor freewheels with the belt system.
Maintenance
Usually there's not a lot involved in the maintenance of an automotive A/C system. Best efficiency can be maintained if the condenser fins are kept clean and clear of debris, and the compressor-belt tension (if adjustable) is proper.
As the system ages, it may develop the smallest of leaks. Should the system be cooling but not at full chill, you may simply be low on refrigerant due to a slow leak. Such very small leaks may be hard to trace, and it may be necessary to add a bit of refrigerant charge about once a year.
You'll want to use caution if you add refrigerant to the system. Follow the specific fill instructions for your setup, always add refrigerant on the low-pressure side, and do not overfill. Careless filling can result in over-pressure and damage to the compressor.
Here you can see the stator that lies behind the clutch rotor. An electrical connector plugs into this to actuate the clutch plates and engage/disengage the compressor operation. The applied voltage creates a magnetic field that pulls the pressure plate and rotor together.
Repair
Leaks can occur at compressor seals, O-rings at the various hoses and components, or the rubber hoses themselves. Gross leaks may be identified by telltale oil seepage at a hose or fitting. Also, fluorescent dies can often be added to the refrigerant and then can be observed later at the point where they leak from the system. Air-conditioning technicians typically also have electronic or ultrasonic leak detectors that can find very small leak sources.
Should any part of the closed system be taken apart, it is necessary to vacuum pump it after reassembly and before recharging with refrigerant.
If the clutch is not engaging to turn on the compressor, the culprit could be due to a number of problems. There could be an electrical harness break, the A/C controller may have failed, or the operation has been disabled by the low-pressure-detection switch in the system. The clutch operation can be quickly checked by using a wire to jump battery voltage to the clutch connector for a few seconds to test if the clutch engages. If it does, then the problem lies somewhere in the wiring or control.
One other thing we should mention involves parts replacement. Should the expensive compressor need to be replaced, the filter/dryer should always be replaced at the same time. One reason is that the dryer loses its desiccant efficiency over time, and compressor failures often leave debris caught in the dryer.
With the rear housing plate of the compressor removed, you can see the valve plate (foreground) and the three pistons in the cylinder block. | 
There are two symmetrical cylinder blocks with three pistons sandwiched inside that ride on a ramp disc on the rotating shaft. The pistons are double-headed to make a total of six cylinder assemblies, three on each side. | 
At each end of the compressor, above the piston tops is a valve plate. Each plate contains three reed-type (one-way) valves. Refrigerant is allowed to enter each piston cylinder where it is pressurized and plumbed to the output port. |