Shotgun 101:
Shotguns are called "SHOT" guns because they throw a pattern of birdshot (little teeny pellets) or buckshot (larger, pistol caliber pellets).
The barrel is smoothbore, like an old musket. Just plain steel, no twist inside to spin anything like a rifle.
Gauge is an old reference to how many lead balls inside the barrel made up a pound of lead I believe. The smaller the number, the larger the gauge.
.410 is the smallest (.45 Colt size), 28 gauge, 20 ga, 16 ga, 12ga, then 10ga is what you normally see.
12ga is the "standard" shotgun gauge for defense and hunting. The smaller gauges are mostly for hunting small birds, rabbits, etc where a 12 would disintegrate them.
For defense, a 00 Buckshot round is 8 or 9 .33 caliber pellets, and the "standard defense load". There's also 000 (six or seven .36 cal), #4 Buck (20 .25 cal pellets) and a few others, but 00 (called "double aught") Buck is the standard by which all are judged.
Shotguns can also shoot slugs. A slug is basically a big hunk of lead the size of the barrel. If I remember right a 12ga slug is .70 or .72 of an inch. They can be surprisingly accurate. My shotgun with rifle sights can hit a silhouette in the head when fired at 100 yards from the prone (laying down) position if I take care. Slugs penetrate like crazy, so they're normally used for hunting, or in a defense situation if the guy's in a car, barricaded behind a wall, etc.
Shotguns come in several basic types.
Single and double barrel: Break-open, load, cock the hammer if it has an external one, and fire. Very simple, design hasn't changed much since the 1870s.
Pump-action. Tube magazine fed, holds anywhere from 3 to 10 rounds, manually operated. Very reliable. You can short-stroke the gun (Not pump it all the way back) so you have to pump it like you mean it. Not normally a problem, but under stress people do weird things.
Semiautomatic: These guns work like most rifles, and the recoil or gas operated action reloads the gun for you, and you just press the trigger again. A bit more complicated for cleaning and using it, but not too much. They generally cost more than a pump action (twice as much, usually). They recoil a bit less.
Specialty types: Lever action (no longer made, or very rare) Winchester made the 1887 lever action shotgun (see it in The Mummy Returns, and T2), but it's been out of production forever. (Since about 1920 I think) though I hear a new one is out. The only other one is the Winchester 1894 in .410 lever action, but that's a squirrel-and-snake gun.
Any other questions?