Open Differential
An open differential is the most common differential used in vehicles. An open differential allows the left and right drive wheels to rotate at different speeds, which is necessary considering that, when turning, the outside tire must spin faster than the inside tire. It's ideal for everyday driving needs, but not for rough, uneven, or low traction terrains. If a vehicle with an open diff were to become stuck in the mud, for example, the tire with the least amount traction would receive full power from the engine while the tire with the most traction would not move. Another problematic example would be a vehicle flexed out on an object, say a large rock, such that one of the rear tires is lifted off of the ground. This tire would spin, while the tire still touching the ground would not. Open differentials are less than ideal for four wheeling, which depends on maximum traction in a variety of terrains.
Limited Slip Differential
The limited slip differential is the OEM's solution to the downfalls of the open differential. Limited slip differentials go by a number of names, most of which are trademarked, and include: positive traction, PosiTraction (GM), Trak-Lok (Dana), Powr-Lok (Dana), Traction-Lok (Ford), Trac-Lok (Jeep), Sure Grip (Chrysler), and TORSEN (Torsen Traction). A limited slip differential allows some difference in wheel speed between the left and right tire, but prevents full power being transmitted to a single tire. This is typically done with a combination of clutch and spring mechanisms inside the differential that, when triggered, lock or nearly lock the left and right axles together so that engine power is delivered to both tires. Power is not usually distributed evenly between the left and right tire, but it's effective enough to greatly increase traction. You'll find that many vehicles are delivered from the factory with a limited slip differential, or some variation thereof.
Automatic Locking Differential
The automatic locking differential builds on the concept of a limited slip differential, but guarantees complete lock up of the left and right drive axles. They also use clutch and/or spring type mechanisms. While the vehicle is coasting or traveling with full traction, the differential remains unlocked. When a low traction situation occurs and at least one tire experiences a reduction in traction, the differential automatically locks the left and right axles together; engine power is then equally distributed between both tires. The concept is great, and the units can handle some serious wheeling. The downfalls is that they tend to engage erratically, even unpredictably at times. While they allow for some turning, you'll often chirp the tires making hard turns because the mechanisms didn't disengage properly and/or locked as a result of the differences in tire speed. But, if you're willing to sacrifice a little tire wear for better traction, there are some great auto lockers on the market.
Selectable Locking Differential
Selectable lockers are the best of both worlds - lock the differential when it's time to go wheeling, and unlock it once you hit the pavement. When unlocked, a selectable locking differential operates like an open one. Selectable lockers are controlled electronically, mechanically, or pneumatically (air). The biggest problems with selectable lockers is that they must use vulnerable wires/lines, which could get tangled up with something on the trail and render them useless. They're also complex and tend to be more expensive, but are the choice of most serious wheelers. Many automakers produce their vehicles with optional electronic locking differentials, although some may not be selectable (actuated by the vehicle's PCM when a low traction condition is identified).
Spool
A spool is essentially a differential delete that permanently locks the left and right axle shafts together. Full power is transmitted to both the left and right tires at all times. Spools are not recommended, and sometimes unsafe for street use, but since there is nothing simpler nor stronger, they are often selected for competition use and dedicated trail rigs. Driving on the road with a spool puts high stress on the drivetrain, and it likely won't last long. The age old "Lincoln Locker" is essentially a poor man's spool. It involves welding the spider gears inside the differential, converting it to a full spool. While this is effective, remember that it is irreversible.