Nothing aka "Open" differential: acts as a normal differential. If one wheel loses traction, it spins while the opposite wheel on the axle stays stationary. Relies on traction control system (brakes) to get out of slippery situations.
Pros: Low Cost
Cons: 100% of engine torque will not be applied to non-slipping wheel. Brakes could overheat if used continuously
Best use: Street trucks that don't see off-road/slippery conditions
Anti-spin aka Limited-Slip: acts as a normal open differential until one wheel is turning significantly faster/slower than the other on the same axle. At that point clutch packs in the differential lock up and both wheels spin at the same speed.
Pros: No user input required
Cons: Cost, clutch packs wear out over time
Best use: Street trucks that occasionally see off-road/slippery conditions or street trucks who want better acceleration under heavy load
Locker: acts as a normal open differential until the user selects locking function. Once locked, both wheels spin at same speed.
Pros: Maximum traction in off-road conditions.
Cons: Cost, unacceptable for street use when locked
Best use: Trucks that frequently see off-road/slippery conditions
For more info on what a differential does, see this youtube video:
Hope this helps!