When choosing compressor you 
should consider if it can constantly provide desired pressure and volume of air. Small airbrush compressors small compressors boast impressive pressure figures but really poor flow rates sufficient only for airbrush pistols. Actually if small compressor can push enough air it will likely overheat in short while.
Our laser cutter was Air Compressor For Laser Cutting china included with small membrane pump (Smallest upon the picture), I really do not know what specifications it has but blast of air was very weak. It is enough if all you have to to do is engrave, although also then we got substantial smoke traces around engraved picture.
it was rated for volume of ~2.4cfm and had a 1 gallon tank which may be filled to 58 PSI. It managed to give us continuous 1 bar (14 PSI) of pressure, nonetheless it had to run constantly and overheat badly within 5 minutes.
Third and current compressor we got was 6cfm rated belt driven unit with 13 gallon container which is filled to ~100psi. With this unit we can run at continuous ~2bar (31psi) without fear of overheating since compressor engine is not on constantly. Using reserve atmosphere in tank we can also press up to 3.5bar (50psi) for brief jobs.
The best setup for me will be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.