Gauge - System of measurement for the internal bore diameter of a smooth-bore firearm based on the diameter of each of that number of spherical lead balls whose total weight is one pound. The internal diameter of a 12 gauge shotgun barrel is therefore equal to the diameter of a lead ball weighing 1/12 pound, which happens to be .729" (Or in British: Bore.)
.410 is not really a gauge, except by convention. It is the actual diameter of the bore in decimal inches.
The Gauge/Bore system is also used to describe the internal barrel diameter of large-bore, 19th century, English, single-shot and double-barrel black powder rifles.