Shot size is the actual size of the pellets in the cartridge. Shotshells use birdshot from #12 up to #1 (the smaller the number the bigger the pellet) and then there are some letter sizes. B, BB, BBB (there are some other sizes for steel shot, but I don't use steel shot, so the heck with 'em). #12 shot is normally called "dust". BBB shot is 0.190" in diameter. After birdshot, you then have buckshot, which goes from #4 (which is 0.240" in diameter) up to 0000 (which is 0.380" in diameter).
There is a #4, #3, #2 and #1 in both birdshot and buckshot, but they are not the same size. If people just use the number (box of #6s, box of #2s), they mean birdshot. If they mean buckshot they say so (#4 buck, #1 buck).
Shotshells are loaded to different power levels, depending on what they are to be used for. You don't need as much power to bust a clay target at 40 feet as you do to knock down a pheasant at 40 yards. They used to list, on the ammo box, the "load equivalent". That meant how the load would do if it was loaded with that amount of black powder. So you would find boxes marked 2 3/4 dram, 3 dram, 3 1/8 dram, 3 1/4 dram, etc. A dram weights 1/16 of an ounce, or 27.3 grains. So a load marked "3 dram equil." would mean that it would be like shooting a load using 82 grains of black powder. They have, pretty much, started to go away from that. They are marking the boxes with the velocity of the charge. They also mark boxes with names that give you some idea of the power of the load.
Target loads are the weakest.
Game load is next.
Then Field load.
Heavy Field, also called Duck and Pheasant load.
The lighter loads not only are slower, but they normally carry a smaller amount of shot. A box of Game Loads might also be marked 3 dr. eq., 1 oz., which means 3 dram equivalent, 1 ounce of shot. A box of Duck and Pheasant loads might be marked 3 1/4,1 1/2, which means 3 1/4 dram equivalent, 1 1/2 ounces of shot.
Also, the more powerful loads normally have bigger shot. Target loads are for shooting clay birds. Normally loaded with 9s, maybe 8s. Game loads are for squirrels, rabbits and quail. Normally 8s, 7 1/2s or 6s. Duck and Pheasant would be loaded with 4s, maybe 2s.
Because they are so much bigger, slugs are weighed in ounces. 437.5 grains to an ounce. A 30 caliber rifle bullet would, normally, be around 150 grains, which is about 1/3 of an ounce. A 12 gauge slug is normally 1 ounce, which is 437.5 grains. I've seen store-bought 12 gauge slugs in 1 ounce and 1 1/8 ounce sizes. I think they even make ounce and a quarter. That would be something close to 600 grains.
There's some good info on shot sizes here. http://en.wikipedia.org/wiki/Shotgun_shell