The shotgun gauge is what most people call "the caliber of a shotgun".

The gauge of a shotgun is related to how many perfectly round balls of the same diameter it's possible to make from one (English) pound of lead. The diameter of one of these balls is the caliber of the gun barrel.

Thus, in a 12 gauge, there's 12 completely round lead balls, each with a diameter of 18.5 millimetres.

This is why a 20 gauge shotgun cartridge is smaller than a 12 gauge: to make 20 equally sized balls, each one must be smaller than a 12 gauge ball.