When choosing compressor you have to consider if it will be able to constantly provide desired Air Compressor For Laser Cutting pressure and volume of air. Little airbrush compressors small compressors boast amazing pressure figures but actually poor flow rates adequate only for airbrush pistols. Also if small compressor can push enough air it will likely overheat in short while.
Our laser cutter was included with little membrane pump (Smallest upon the picture), I do not know very well what specifications it has but blast of air was extremely weak. It is sufficient if all you want to do is engrave, although actually then we got substantial smoke traces around engraved picture.
it was rated for level of ~2.4cfm and had a 1 gallon container which can 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 five minutes.
Third and current compressor we got was 6cfm rated belt driven unit with 13 gallon tank which is filled to ~100psi. With this unit we can run at constant ~2bar (31psi) without concern with overheating since compressor engine is not on all the time. Using reserve air in tank we can also drive up to 3.5bar (50psi) for brief jobs.
The best setup in my opinion will be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.