LSD is a mechanism that typically uses clutch plates to ensure that after a "bit" of spinning, engine torque is sent to the motionless wheel. Providing necessary slippage in turns, an LSD provides excellent street manners.
The two downsides to this are the "bit" of slippage is too much for some off road applications. Secondly, as the clutch plates wear this slippage increases to a point where it is no longer functional and becomes an open diff.
An LSD is a great factory option if one intends to travel off pavement only occasionally or is frequently in snowy or icy conditions.
To tell if you have a Limited Slip lift the end of the vehicle and place a couple of jack stands underneath. Place the transmission in neutral. Rotate one wheel. If the other wheel moves in the same direction you have limited slip. If it moves in the opposite direction you have an open diff.
Auburn Gear
Thorsen
Trac Lok
Made by Dana Spicer (two spider gears transmit torque through assembly).
Power-Lok
Made by Dana Spicer this gear driven LSD is one of the most positive, durable and aggressive units available (four spider gears - much stronger than trac-lok).
Detroit TruTrac
This limited slip differential is gear-driven and very robust. It requires no lubricant additives.
An automatic locker is a step above an LSD for off-roading. Replacing the clutch plates are more robust mechanisms that lock the two front axles together, almost constantly. Some release is required to prevent the axles from turning into popcorn around corners, but this is much less than in an LSD. This release is what makes this an "auto" locker. Full lockers do not release at all.
The obvious advantage is that the installation of both front and rear lockers will provide constant true 4WD.
The disadvantages are fairly significant and are why this is rarely a manufacturer's option (Land Cruisers and Hummers are two exceptions). Drivability is negatively impacted especially in snowy conditions. Lockers can plow straight ahead in tight or low traction turns. Many lockers make additional noise which some may find objectionable. Tire wear is also increased significantly.
If you have manual locking front hubs some of these disadvantages are greatly reduced.
The lockers below are listed by cost, least expensive first.
Welded/Spool
For the best in off-road performance some weld their axles together. This option is available only for those rigs that are never driven on the street. This and the ARB below are the only full lockers listed here.
Detroit EZ-locker
It's low cost and the fact that the average shade tree mechanic can install this without expensive tools makes it very attractive. It installs into the original open differential case so no gear set-up is needed. This unit provides excellent traction off-road.
Lockright
Inexpensive and can be installed by a DIY mechanic as it also uses the original differential case. The Lockright is noisier than the Detroit Locker. This unit provides excellent traction off-road.
Detroit Gearless
A new locker advertised to be almost as strong, quieter and better behaved than the Detroit Locker. At this writing no experience is available in the group. This is also reputed to be a DIY job. Requires Detroit Silencer fluid.
Detroit Locker
The Detroit Locker is the preferred choice for serious no-compromise off-road work. Known for its strength it also quieter than the Lockright but makes a LOUD pop when releasing. Some have gotten used to this and even say they enjoy it. No special lubricants are required.
ARB
This is the only widely available locker that can be disengaged. This allows for the unparalleled off road capabilities of a full locker and when disengaged, none of the drawbacks. In simple terms at the driver's discretion an air compressor will force the axle's locker mechanism into place providing "welded axles at the flip of a switch".
Note that this provides true and complete locking of the axles and therefore puts more strain on components than the other "Automatic" lockers described here.
The two reasons why we all don't have ARB lockers is that they are significantly more expensive and pose reliability issues associated with the air lines, compressor and electrical bits.
This is defined as the relationship of the angle of the driveshaft to the yoke of the rear axle. To reduce vibration and wear for extreme lifts the pumpkin can be tilted to line up with the driveshaft with shims under it, or a re-welding of the spring perches. The more common and less desirable approach is to lower the transfer case.
There are many gremlins that can cause drive train vibration. An excellent troubleshooting chart can be found at Performance Unlimited.