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Copper wire is made by drawing copper rods through dies of decreasing sizes. More dies are needed to make a thinner wire. The American Wire Gauge (AWG) number is the actual number of dies used to make the wire. A 24 AWG wire is made by drawing the copper through 24 dies.

A thinner wire of the same material has a higher resistance per unit distance than a thicker one. In house wiring, a thicker gauge wire (lower number) can usually carry a higher current. 14 gauge Romex cable usually has a 15 Amp breaker and a 12 gauge cable has a 20 Amp breaker. This assumes that the switches, outlets, etc. have the same current rating as the cable. Putting a 20 amp breaker on a 14 gauge circuit produces a potential fire hazard. As I am not a licensed electrician and codes and circumstances vary, this paragraph is for informational purposes only.