mjoeking, a couple of other points.
Single wheeler diffs have simple gear trains inside them that apply equal torque to both wheels (as has been previously described by others). If one wheel has no torque available (e.g. on ice), than that's ALL the tractive effort that's available...that wheel will spin, and power = torque x rpm, more POWER will go to the spinner, and you won't go very far/fast. Easily identified by jacking both wheels off the ground, and spinning a wheel...the other one will go backwards.
Limited Slip has (as has been described by others), a clutch of some sort....many have packs of clutches, they used often to be metallic cones (cone type LSD). These clutches are usually between the differential side gears and the rotating diff centre. They CAN be one side only, but usually symetrical. Being between the side gears and centre casing, the more load that is applied, the more squeeze is put on the clutch packs (or cones), making the diff tighter. Identifiable by jacking both wheels up, and turning one...the other will go the same direction...can also be identified by squeels and chirping as owners drive around concrete carparks, or painted lines.
Lockers...aren't truly a differential, but a clutch arrangement. A true differential, the centre will turn at the average speed of the wheels, a locker, the centre turnes only at the speed of the driven wheel. Due to (dog type, not frictional) clutching arrangements, the unloaded wheel will over-run the speed of the centre, and only provide lock up when the drive wheel speed matches the over-running speed, i.e the wheel slips...clunky, but effective, can transfer nearly 100% power to the wheel whose axle you are about to break.
Spool (or CIG/welded locker), fixed, so that there's no differentialising action...can't do U turns.
Viscous LSD uses a viscous clutch instead of a traditional clutch...relied on the unloaded wheel spinning, and the difference in speed to apply torque to the planted wheel (not a fan, but they are pretty OK in a FWD as I found when my parents owned one).
Torsen...these are really, really clever. Think of a worm gear, and how you can easily drive a ring gear with a worm, and can't drive the worm gear via the ring gear. That is called "non backdriving", you can't reverse the power flow. get the angles right, and you can create a "backdriving worm gear"...couple a backdriving and a non backdriving gear in a differential centre, and you can create a true differential where you can turn one wheel and have the other go backwards, but will not allow a wheel to massively increase in speed, and so biases torque to the loaded wheel, all through these worm gears. Through angle design, you can vary the torque split.
There's others these days that use electromagnetic fluids controlled electronically to provide torque split, internal electronic clutches, and lord knows what.