Known for its hunting and clay busting prowess, the Over/ Under simply gets its name from the barrels mounted over and under each other. Almost exclusively used in Olympic competitions.
The gun most traditionally used in film and TV, the 'Pump action' shotgun. The cartridges are 'cycled' or kicked out of the chamber by the pumping action of manually pulling back and then pushing forward the wooden forend.
The semi automatic or 'semi auto' is well known for use as a 'wild fowling' gun ie hunting of birds such as geese and ducks. It is also used for clay pigeon and rough shooting. These hold from 3 to 12 max cartridges. The shells are automatically ejected by the gun and load the next cartridge automatically.
The traditional style of side by side getting it's name from the mounting of the barrels side by side. Earlier versions were also called hammer guns due top the hammer style of the firing pins. Not much in vogue these days, fine English collectable examples can fetch hundreds of thousands of your finest British pounds.
Usually smaller calibre, single barrel 'folding' guns are one shot. The main uses are either close up target shooting of vermin and pest control. The main pivot screw in the middle of the gun allows the gun to nearly fold in half for better storage and carrying purposes.
Tactical shotguns get their name from the variations of the predominantly pump and semi auto styles of shotguns, used for military and security purposes. These would have attachment rails called picatinny rails for mounting accessories such as laser sights or torches. They would be made form much hardier composite materials.