The standard differential is a completely open differential. When you're going straight on a hard surface, it supplies equal traction to both tires. When you turn, the inside tire slows down. This is good for on-road driving, but if you're stuck, an open differential allows one tire to spin freely while the other is stuck.

A limited slip differential is built to alleviate some of these problems. Like the name implies, it limits the slip by sending some of the torque to the wheel that has the most traction. The Jeep Trac-Lok differential is an example of a limited slip differential. Without getting too much into details, a limited slip differential uses friction between the differential case and a series of clutch plates to achieve this goal. A limited slip differential provides civilized on-road performance and reasonable traction in off-road situations, but it may accelerate tire wear and require special lubricant for the differential.

A locking differential is useful for really complicated off-road driving situations, like rock crawling or lots of mud. An automatically locking differential works using a clutch that is engaged most of the time, but disengages when one wheel is traveling faster than the other (like when turning). This provides the best traction for off-road driving, but can cause squirrely handling for on-road driving, especially in slippery conditions (such as icy or snowy roads). Certain lockers are more forgiving than others, so it pays to check out reviews and ask people's opinions.

Finally, there are on-demand locking differentials. The best examples of these are the ARB Air Locker (an aftermarket item) and the Tru-Lok differential (standard on the Rubicon model). When they are disengaged, they behave like an open differential (or a limited slip, in the case of the Rubicon). The locking is actuated by a pushbutton on the dashboard. Then, the locker is engaged via an air hose or by other means. These are the most reliable lockers for mixing both civilized on-road driving and good traction for off-road driving. Unfortunately, they're also the most expensive.

This article from Jeepforum.com discusses differentials in more detail.