If you want the ability to negotiate difficult rapids, a white-water kayak is your best bet for river kayaking. White-water kayaks have a short body and a lot of "rocker" or curvature between bow and stern. This allows them to turn extremely quickly to avoid rocks and negotiate rapids. The downside is that white-water kayaks are slow and track horribly. They have no natural tendency to stay in a straight line.
At the other extreme are long, sleek sea kayaks. These kayaks are fast and track extremely well. Many also come with storage compartments, making them a good choice for both casual paddles and long trips. In general, the longer and thinner a sea kayak, the faster it will be, but the more sluggishly it will maneuver. There are many variations in sea kayak design, and some allow for quicker turns for slightly rowdier rivers.
For casual river kayaking, sit-on-top kayaks are a smart option. Instead of sitting in an enclosed compartment, the kayaker sits on top of an open cockpit. Many people find this more comfortable, since it is easy to move around. Sit-on-top kayaks are also stable, which makes them appealing to people who find normal kayaks a bit too tippy. They also work for river fishing, since there is room in the cockpit for your gear and a lunch.
Recreational kayaks make good, all-purpose river kayaks. They are a bit more rounded and less sleek than ocean kayaks, but they track better than white-water kayaks. They aren't very well suited for long trips or extreme white water, but are good for nearly all other conditions. If you aren't sure what you want, a recreational kayak may be the ideal first boat.