Pump guns where the "barrel bends down" are spring piston guns. There are several type of these guns, break barrel, side lever, or under lever, occasionally a different novel method. They all cock a powerful spring, which pushes a piston, which supplies the compressed air. Spring guns can have their own little quirks, such as recoil, "barrel droop", inconsistency, dieseling, hold sensitivity, etc. They can also be very powerful and are self contained. The good ones are from Europe and can be rather expensive. I generally find these fastest of the single shot airguns to shoot and reload. These also have a potential hazard, the possibility that the lever slips while cocking or loading a pellet, causing the breech or barrel so slam closed on one's fingers. This can cause lost digits. Most spring guns have some kind of safety system designed to prevent such an incident, but only a fool would rely on such a device. It's wiser to firmly hold the cocking lever/barrel while loading in the event of a failure to prevent the piston from closing unintentionally. I used to wonder what the chances were of this happening, and it happened to me once. Thankfully, I had the cocking lever firmly in hand and my fingers were out of the breech. Really scared the crap out of me.
Pre charged pneumatic (PCP) guns can be very consistent, usually little or no recoil, very accurate, very expensive, very powerful, offered in widest range of calibers. They need expensive support gear because of the rather extreme air pressures involved, either scuba tanks or a mechanical pump that is no fun at all to use.
Manual pump pneumatic guns can be very consistent, usually little or no recoil, can be very accurate, but are on the low end of the scale. Daisy, Crosman and Benjamin/Sheridan are the big names. They generally are good plinkers, the higher end Benjamins and Sheridans can be very good small game hunters, particularly in .20 and .22 calibers. The downside of these guns is that they generally need multiple pumps (usually 2 - 10)for each shot, which can be tiresome. Daisys and Crosmans have lots of plastic, and have limited choices in the larger calibers. Benjamins and Sheridans have some quirks such as being a real pain to scope.
There are also CO2 powered airguns. These tend to be Dasiy/Crosman low end guns, but there is a very healthy modding community with Crosmans for squeezing the most accuracy, power, and CO2 out of these things. The .22 CO2 guns are good for hunting. Little or no recoil. On the downside, CO2 does poorly in cold weather, and can be inconsistent as the gas is depleted from the cartridge/tank.
Most airguns are single shots. There are occasionally repeaters, semiautos, DAO repeater guns. These tend to be on the low end, but not all.
Pistols come in a variety of configurations, too. Some, such as Crosmans (multi pump or CO2), can be ok or small game hunting, particularly in .22.
Each style has it's own pluses and minuses. I like them because I can shoot every day here in my big city back yard without ever bothering anyone or schlepping out to the range or BLM land.
