What Size Wire for an Air Compressor?

It depends on whether it plugs in or is hardwired. A small 120V portable or pancake compressor (up to about 1.5 running HP) is cord-and-plug, so you do not design a circuit for it: plug it into an existing 15A or 20A branch circuit, and the branch breaker plus the compressor's built-in thermal overload protect it (NEC 430.32(C)). A hardwired stationary shop compressor is a motor load under NEC Article 430: the conductor carries 125% of the table full-load current (430.22) and an inverse-time breaker may be up to 250% of it (430.52). A 2 HP 240V unit lands on a 30A breaker; a 5 HP 240V unit on 70A with #8 copper.

No. Air compressor horsepower is notoriously overstated: a 'peak' or 'developed' 5 HP marketing figure often maps to a 2-3 running-HP motor that draws far less. Size the circuit from the nameplate running amps (or the true rated HP), never the peak HP on the box. The NEC uses the table full-load current for the motor's actual rated horsepower, so if the running amps and the advertised HP disagree, the nameplate amps win.

Usually yes for a small 120V unit, with two cautions. First, a compressor draws heavy starting current, so it should be the only significant load on that circuit and ideally on its own 20A receptacle; running it on a shared 15A kitchen or lighting circuit invites nuisance trips. Second, a long or undersized extension cord causes voltage drop that overheats the motor, so use a short, heavy-gauge cord (or none). Larger 240V compressors are hardwired or use a dedicated NEMA 6-20/6-30/6-50 receptacle sized to the circuit.

Because a motor circuit protects the wire and provides starting headroom with two different devices. The branch-circuit breaker (NEC 430.52) is sized up to 250% of the motor's full-load current so it does not trip on the inrush at startup; it only handles short circuits and ground faults. The conductor is protected from overload by a separate overload device set at 115-125% of the nameplate FLA (430.32), which on most compressors is the built-in motor thermal protector or pressure-switch overload. NEC 240.4(G) exempts motor conductors from the small-conductor breaker limits, so a small wire on a large motor breaker is code-correct.