Is it "to reload" or "to shoot"? I figured out that "to pump bullets" means "to shoot", but I don't know if there isn't a difference with shotgun.
Some of the other answers are pretty close, but still not quite right. Pumping a shotgun doesn't reload it, and it does more than eject the spent shell casing. When you pump a pump-action shotgun, you are cycling the action - that means you are doing whatever the gun is designed to do automatically from a single movement.
This can all be pretty confusing to a person unfamiliar with guns, whether or not you are a native English speaker!
The firing of a gun consists of a few steps. First, you have to get a round into the chamber - this is called "chambering a round." The chamber is the part of the gun that holds the cartridge in place prior to firing it. This might involve removing a spent cartridge if it has not already been ejected. Then you have to close the bolt, which makes sure propellant gases and the cartridge can't escape from the open chamber when you fire the round. Next you have to cock the hammer. The hammer is the part of the gun that is under spring tension and is released when you pull the trigger to hit the firing pin. After all that is done, you can finally fire!
The "action" of a gun is all of the parts of that process that happen automatically in response to one action. For a pump-action shotgun, that includes opening the bolt, ejecting the spent cartridge, chambering a new round from the magazine, and cocking the hammer.
"Reloading" usually refers to inserting rounds into the magazine so that they can later be chambered by the pump action. You can put a round directly in the chamber, and that might technically be called a reload, but in that case you wouldn't operate the pump action until after you had fired the round.
"Shooting" or "firing" is the act of depressing the trigger, causing the hammer to strike the firing pin which in turn strikes the primer and ignites the propellant in the cartridge.
Other types of guns have different actions. There are pump-action rifles which operate in basically the same manner as pump-action shotguns, but there are also bolt-action, lever-action, falling-block action, break-action, and gas-operated and recoil-operated weapons. Revolvers can have single- or double-actions - in a single action revolver, the next chamber rotates into the firing position when you pull back the hammer, and then you pull the trigger to fire. In a double-action revolver, the hammer is cocked and the cylinder rotates as you pull the trigger.
In all of these cases, the "action" is the part of the process that the gun does for you. Cycling the action sometimes has a common name based on what the most obvious movement is - pumping in this case.
It comes from a pump action shotgun or rifle.
A pump-action rifle or shotgun is one in which the handgrip can be pumped back and forth in order to eject a spent round of ammunition and to chamber a fresh one.
Here is a tutorial on shotgun pumping. So you're not actually loading the shotgun, you're ejecting the spent round when you pump the shotgun. You can imagine that if you run out of rounds, you can pump all you want, but the shotgun will not be loaded.
Here, the person is loading/reloading the shotgun.
Pumping a shotgun means to reload. The action of pumping discards the spent shell and loads a new one in the chamber.
Yes, there is a difference as shotguns don't have bullets to pump. It's conceptually different either way.