Before explaining how the limited-slip action works, let's look at how the differential part works.
When you go around a corner, the inside wheels have less ground to cover than the outside wheels, so naturally, they spin more slowly. The differential's job is simply to allow the two drive wheels to turn at different speeds while still staying mechanically connected to the gearbox.
When the car is going straight, the differential itself spins at the same speed as the wheels, and from your point of view as an honorary differential, all the gears inside are stationary. When the car goes around a corner and one wheel has to go faster than the other, all you'd see, as a differential, is the two axles turning in opposite directions.
Now look at the diagram of the Quaife differential and imagine the sun gear in front (the gears attached to the axles are called sun gears) turning clockwise. The pinion gears around the outside of the sun gear therefore turn counterclockwise. The pinion gears from one side of the differential mesh with pinion gears from the other side, so they turn clockwise, and those pinion gears turn the sun gear on the far side counterclockwise. That, if you've lost track, means it's turning in the opposite direction from the sun gear in front, which is just what you, as a differential, want to see when you're going around a corner.
Coasting around that corner, there's no load transmitted through any of these gears, but hit the throttle and things get complicated. Look closely at what happens when the differential tries to drive the wheels. Assuming the car is going straight and there's equal grip on both wheels, the differential housing turns, which forces the pinion gears, trapped in little pockets around the perimeter of the diff, to move with it. These pinion gears are meshed with the sun gears, and because none of the gears are turning (relative to the diff housing, that is), the sun gears have to follow along, spinning at the same speed as the differential.
The gear teeth can only push on each other with a force directly perpendicular to the face of the tooth, so if the tooth is angled, so is the force on the gear teeth.
Now, the pinion gears don't rotate on a shaft; instead, they sit in tight-fitting pockets, with the tips of their gear teeth rubbing on the inner walls of each pocket. When the differential housing turns and the pinion gears push on the sun gears, the pinion gears get shoved back against the walls of their pockets. Of course, the more torque that gets applied, the harder they get shoved against the wall.
That's going to be very important in a second.
When one wheel loses grip and tries to spin, all the gears have to start turning, just as they did when we were coasting around a corner, but now that the pinion gears are being shoved against the walls of their pockets, there's some resistance. This is just the beginning of the resistance, though. In the middle, where the pinion gears mesh and the pinion on the gripping side tries to turn the pinion on the slipping side, something interesting happens.
The pinion gears are cut with helical gear teeth that mesh at an angle. When these angled teeth push against each other, the angle of the teeth causes them to pull each other up against the end of their pockets.
The combined friction of the tip of the pinion gear teeth rubbing against the pocket walls and the ends of the pinions rubbing against the ends of the pockets actually creates enough resistance to prevent the inside tire from spinning.
When designing a new application, engineers can adjust the amount of resistance in the diff by changing the shape of the gear teeth and the angle of the helical pinion gears. To see how this works, just look closely at the interface between two gear teeth, starting with gear tooth No. 1 on the left. The bottom tooth is pushing up with a force represented by the little green arrow. At the gear tooth interface, the bottom tooth pushes up on the top tooth with that same force (but now the arrow is white), and if there's enough resistance on the other gear, the top tooth will push back with the same force.
The gear teeth can only push on each other with a force directly perpendicular to the face of the tooth, so if the tooth is angled, so is the force on the gear teeth (the white arrow). In gear tooth No. 2, you can see the white arrow is now angled. This same force can be looked at as two forces, one pushing up (green), and one pushing to the side (red). As the angle of the teeth gets steeper, as in gear tooth No. 3, the amount of force pushing to the side increases. The force shoving the pinion gears into the pocket wall comes, in part, from these little red arrows.
The majority of the resistance and virtually all of the tuning happens when the angle of the helical pinion gear teeth is changed. A steeper helix translates directly into bigger red arrows shoving the pinion gears into the end of the pockets. If there's no resistance from the gear on top, the gears will simply turn and there won't be any red arrows at all. No red arrows mean no limited slip, so if one wheel is completely off the ground, the diff will act as an open differential.