Compressors have three main components: a motor, the compressor assemply, and a stroage tank. The motor provides the power to the compressor, which does the work. The tank stores the compressed air for use.
Oil vs oilless
Compressors that use oil are less noisy and last longer than oilless compressors. They do add some of the oil to the air they produce. Oilless compressors are cleaner and easier to use. Portable units for use in finished spaces are typically oilless to eliminate spraying oil in living spaces.
Pick an oilless compressor for indoor use. Go with an oil model in all other cases.
Screw vs piston compressors
Screw type compressor have a much longer service live due to the smoother operation of the spinning screw. They are exclusively found on industrial compressors. Well built piston compressors offer prefectly reasonable serive lives and should not be discounted.
Single vs two-stage
Multi-stage compressors produce a smoother air-flow and last longer. Single stage models are perfectly usable, if sized correctly and combined with the correct tank.
1) Choose a CFM number
The root of the decision making process is sizing the air compressor correctly. Find the air requirements for the largest tool you want to run. That number is cfm of air the compressor has to supply to satisfy your needs. It's prudent to add 20% to this number to allow for some headroom. If you are running multiple tools at the same time, add all numbers together.
2) Choose a tank size
Next consider the tank size. Large tanks buffer air for later use. The allow a smaller pump to temporarily perform like a larger one until the tank runs out. Emptying the tank and running the tool directly from the compressor only works marginally. Piston type compressors produce waves of pressures, which the tank normalizes. Size the compressor such that the tank never truely empties during operation.
3) Check the duty-cycle of the compressor
Most compressors are not design for 100% duty-cycles. They need to cool down for a while after filling the tank. Most users find out about this problem when they try to use a spray gun with small portable compressor. While the pump is strong enough to supply the gun at the begining, it will soon overheat and shut down. This is simply a design limitation and not a fault of the tool. Choose the correct compressor for your application.