A LSD is a thing to control traction.
normally as you go round a corner the outside wheel is required to move faster than the inside wheel. If you had a solid diff this would not happen and the car would try to go straight ahead and you would litterally have to drag it around the corner.
As a result a diff lets the power be shared between the two wheels whilst still allowing the outside whell when cornering to move faster.
All this is fine until you encounter a loss of traction, in this case the normal diff will transfer all power to the wheel that is slipping and none to the wheel that has maintained grip. This is bad news particularly if you are a 4wd you just sit there spinning your wheels uselessly.
So a limited slip diff is used to provide some amount of difference in wheel speeds but it will then transfer power to the wheel that maintains traction.
The reason this is useful in racing is going round corners on two wheels means the inside wheel has no traction

OK this doesn't happen all the time but it does mean that the inside wheel is more inclined to slip. the LSD prevents all power going to that wheel and hence a loss of forward momentum.
The phantom LSD works by packing the middle of the diff with two plates seperated by some pretty hefty springs which force the plates against the cogs(for want of knowing the correct term) at either end of the diff this provides a resistance to the two wheels moving at different speeds.
There are also different types of LSD
Viscous couplings
Clutch pack types
The one they use in the Hummer (Torsen?)
and some more as well
Apparently the Skyline one is quite innovative it allowed the car to behave like a rear wheel drive (oversteer tendencies) but when the LSD cut it it let you have the traction to drive out of the corner.
Now with traction control a computer monitors the speed of each wheel and if a wheel is more than some tolerance faster than the rest the computer applies braking force to that wheel which distributes the power to the other wheels.
Oh Apparently the early racing minis used a solid diff so that gave them more traction but they really had to man handle the cars around the track. Minis being lighter than most cars this wasn't the problem it would have been with a heavier car.
The following link can explain it far better than me though
http://auto.howstuffworks.com/differential.htm
Will a LSD help to control torque steer

I would have though it would go towards helping