There are essentially two major categories of commercial chilling technologies: compression and absorption.

There are three types of compressors used in compression chillers: centrifugal, rotary and reciprocating.
Centrifugal compressors:
The majority of centrifugal chillers are water-cooled and it is often used for medium to large cooling loads from 200 to 6000 tons of refrigeration. you will rarely find a centrifugal compressor in air-cooled chillers.
Centrifugal compressors are usually driven with electric motors, but it is also possible to drive chillers directly with reciprocating engines, combustion or steam turbines.
What about the working principle of centrifugal compressors?
Like centrifugal pumps, an impeller provides the force to compress the refrigerant vapor. The low-pressure refrigerant vapor enters the eye of the impeller. The rotating impeller adds kinetic energy to the flow which is then converted to an increase in pressure by slowing the flow through a diffuser.
Reciprocating compressors:
Reciprocating compressors operates similar to a car engine, it uses a piston driven from a crankshaft. The refrigerant is drawn into the cylinder during the downstroke and compressed in the upstroke.
These compressors could be found on air-cooled and water-cooled chillers up to 200 tons of refrigeration.
The third type is the rotary compressors, which could use one of the following:
Scrolls.
Rotating vanes.
or, Helical screw.
The helical screw is the more common type, they are used on both air and water-cooled chillers. These chillers are typically available in 70 to 400 tons of refrigeration.
Screw compressors work by using two interlocking rotating helical rotors to compress the refrigerant.

Absorption chillers:
Absorption cycle uses heat to generate cooling using two media:
A refrigerant and an absorbent. Water/lithium-bromide is the most common refrigerant/absorbent media pair, but other pairs can be used.
The absorption process uses an absorber, generator, pump and recuperative heat exchanger to replace the compressor in the vapor-compression cycle.

Compression chiller working principle:
https://youtu.be/4nNW0aSEIyA