Pump paintball guns are the original. When the sport was invented, all that there were was pump guns, used primarily for marking trees when surveying, running on CO2 cartridges. Over time, advances in the paintball marker market mean that pump markers have lost ground to faster and fancier electronic guns, which have allowed for entirely new kinds of paintball games. But that doesn't mean there aren't still some players who prefer the pacing and the almost personal nature of a pump game.
If you're new to pump guns, then first you need to understand what sets them apart from other paintball markers. Pump guns work like an old single-action revolver, in that cocking the gun loads the next round, or in this case, paintball, and readies the gun to fire that shot. The term pump gun can be misleading, since they still use a propellant, often CO2 cartridges.
Pump paintball can't utilize the same tactics as faster-paced variants like speedball, as a pump marker can't fire quickly enough to suppress a target or support a pray and spray style of play. This leads to a slower-paced, more cerebral paintball game, removing the advantages of higher-priced gear and replacing them with a match where you play the player, not their gear.