Gravity-fed hoppers perform as the name implies. It uses the force of gravity to feed balls to the marker. Prone to chopping paintballs, it is the simplest, and cheapest, hopper available. And gravity never fails on field (unless you're upside down).
Agitated hoppers are a slightly more advanced design, they were created to rectify the many concerns of gravity fed hoppers. There is a small vibrating mechanism that, as the name implies, agitates the paintballs. This greatly reduces chopping, but requires batteries.
Force-fed hoppers often employ the use of an impeller to force paintballs into the marker, making for some impressive feed-rate times compared to either the other types. It also allows users to modify the traditional hopper-over-the-barrel placement to other places, such as under the barrel as seen in the Q-loader, lowering gun profile.
Cyclone feed hoppers are hoppers designed just for a cyclone feed system. They are gravity fed, but because they use a cyclone feed system (which is built into the marker) they can crank out 15 bps (ball per second)