Pump air pistols are the most powerful, most accurate, and generally the most expensive of the air pistols. However, pump air pistols, also called pneumatic (powered by compressed air), are great for target shooting, small game hunting, and can even be used for self defense, though a 38 caliber or higher pistol would have more stopping power. A 357 magnum revolver or 9 mm semi-automatic pistol are great self defense guns.
Pump air pistols have a level, usually under the barrel, which is pumped a number of times to build up air pressure, for the desired power level. Multi-stroke pneumatic pump air pistols can be pumped up to 10 times before single shot, or could be shot after a pump or two with less power. At full power, a 22 pump pistol might approximate the power of a 22 pistol, with similar accuracy. A number of good brands have been around for years. Benjamin and Sheridan have made quality pump air pistols for decades. Crosman pump air pistols have also been around for a long time.
There are also single stroke pneumatic pump air pistols, but these are generally less powerful than multi-stroke, though quicker between shots.