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Larger shot numbers mean smaller pellets:
The diameter of birdshot is equal to .17" minus the shot number; #4 shot is .13" in diameter.
Same goes for buckshot; 000 is .35", 00 is .33", #4 buck is .25".
Larger shot penetrates deeper, so it is used on larger animals.
Small critters: doves, squirrels, or the clay pigeons used in competition, can be taken with shot as small at #8. Some people like to use larger shot because it can kill at slightly longer range and reduces the number of pellets you have to remove from your food.
Larger critters, such as pheasants, turkeys, or rabbits should be taken with slightly larger shot such as #6 or #4.
Even larger critters, such as ducks or geese, require larger shot, compounded by the use of steel, #4-BB is recommended for for ducks, #1-TT (TT = .21") for geese.
The largest critters that buckshot is suited for, such as deer, coyotes, men, or perhaps already wounded lions, should be shot with pellets no smaller than #4 buck.
Steel shot is less dense than lead, so it loses speed faster in air and penetrates less in tissue. Larger shot (rule-of-thum

2 birdshot sizes bigger) is required to get performance similar to lead.
The pattern - how and how much the load spreads out during flight - of a load of shot changes depending upon the existence of a choke in the barrel. The choke does two things:
Dents lead that passes through it. This is usually considered a bad thing, as it makes the pellets disperse more - and disperse erratically, making for uneven patterns.
The tighter the choke, the more the lead will be dented.
The more lead in a shotgun load, the more serious this effect is.
The softer the lead in a shotgun load, the worse the pellets will be damaged.
Buffered shotgun shells pack padding around the lead pellets to protect them. This results in tighter patterns.
Helps make all the pellets stay in one cloud. Tighter chokes do this to a higher degree. Yes, this is opposite to the previous affect.
Most chokes anymore are interchangeable; the inside of the end of the barrel is threaded, and chokes can be screwed in and out at will. Generally, the following chokes are the most common:
Cylinder bore, which has no restriction, throws open but very uniform patterns; it is useful for close-range shooting
Skeet choke
Improved cylinder
Light Modified choke
Modified choke
Improved Modified choke
Light Full choke
Full choke, which has a lot of restriction, throws very tight patterns when fed quality shot. However, when fed soft lead, a full choke tends to shoot awful patterns: uneven and possibly larger than those produced by a looser choke.
Slugs, as mentioned, are large pieces of lead propelled by powder, loaded one per shell.
Slug power is determined in the same way as buck shot, with standard, magnum, and super-magnum loadings. With magnum and super-magnum are often loaded with heavier slugs.
The "rifled" slug, used in smooth-bore shotguns (most) has rifling or ribs on the slug. These ribs are designed to prevent damage to the gun if fired through a barrel with some choke.
A properly constructed (durable and made of hard lead) 12 gauge slug essentially transforms a shotgun into a .73 caliber rifle capable of taking all game animals on the North American Continent, though the range is somewhat limited.
"rifled" slugs don't work well in rifled barrels.
Sabot slugs are for rifled barrels.
A sabot is a covering for an object which is smaller than the bore of which it travels through. That is, sabot slugs "wear" a plastic jacket that is discarded after they leave the barrel.
Sabot slugs are generally more accurate, higher velocity, and more aerodynamic than "rifled" slugs, but they require a rifled barrel to be fired accurately.
Both types of slugs are fairly low velocity, and thus have rainbow-shaped trajectories that are not helpful to long-range shooting. It is fair to say that "rifled" slugs can reach out to 100 yards without too much difficulty; sabot slugs about 150 yards.
Note that generally buck shot and slugs in the same size as bird shot will have much more recoil than bird shot. So, the first time firing slugs or buck shot, be prepared for extra kick (if you use magnums or super-magnums, be prepared for a lot of extra kick).