Air compressor starts and then shuts off

Do you mean it runs for 1 or 2 seconds and stops or it runs for minutes and stops? If it never really starts (runs for a second or less) then it is probably a capacitor start motor and the capacity has gone bad.

If it runs for minutes or more and then stops and subsequently restarts when it has cooled then the thermal overload switch is turning it off when the motor overheats. If the unit is not all that old, hasn't been run particularly hard and you have this problem every time you turn it on when it is just pumping up, (which is to say, there is no good reason for the motor to be overheated) then the thermal overheat switch is probably defective.

If the compressor is being run hard when this happens, then the unit is being overheated from heavier use than for which it was designed. If the interior of the compressor motor has a coating of saw dust or dust, it acts like an insulator and results in the motor heating more quickly than it should under normal circumstances. Also, if it is being used in a hot environment it will likewise have trouble cooling.

So, if it really doesn't run at all, it is probably the motor starting capacitor. If it happens when the compressor isn't really hot, then the thermal over-temperature switch is probably defective. If it gets genuinely hot, but without being run hard, then the motor might need to be cleaned because it is not cooling properly. If it happens when it is hot and it is hot because it has been run hard, then either don't run it as hard, cool it better or get a bigger compressor. If it is clean inside, not being run hard but still overheats, then I would look at a malfunction in the compressor portion of the unit having more friction than it should (bearings lubricated?) thus requiring more horsepower to run the compressor than the motor is designed to generate and results in overheating. Low voltage can also do the same.