When choosing compressor you need 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 impressive pressure figures but really poor flow rates adequate limited to airbrush pistols. Also if small compressor would be able to push enough air it will likely overheat in couple of minutes.
Our laser cutter was included with little membrane pump (Smallest on the picture), I do not know very well what specifications it has but stream of air was very weak. It is adequate if all you want to accomplish is engrave, although even then we got substantial smoke traces around engraved picture.
it was rated for volume of ~2.4cfm and had a 1 gallon container which may be filled to 58 PSI. It were able to give us constant 1 bar (14 PSI) of pressure, nonetheless it had to operate all the time and overheat badly within 5 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 continuous ~2bar (31psi) without concern with overheating since compressor engine is not on continuously. Using reserve air in tank we can also force up to 
3.5bar (50psi) for short jobs.
The best setup in my opinion would be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.