First of all, differentials directly transfer
torque, not power (although they are related).
There are four kind of differentials kinds of differentals that come to mind:
Open: This type offers almost no resistance to delivering different torque amounts to the opposite wheels. When load is placed on, say, the left wheel, all of the torque is delivered to the right wheel--causing undesirable spin. All LSDs get around this by forcing some of the torque to be delivered to the other wheel. This is expressed as a ratio, such as 4:1, which means that for every 4 ft-lbs of torque going to one wheel, 1 ft-lb will be delivered to the other wheel.
Clutch LSD: This uses a pair of clutches and spring on the bevel gears at the inboard side of the driveshaft to tie them together. The stiffness of the spring determines the torque bias ratio. Some "active" differentails use a modified version of this system where the clutch pressure is controlled by a fluid pressure which can be dynamically changed.
Helical/Torsen LSD: This type of differental uses helicals gears. When torque is applied to one wheel, the sideways forces on the gears causes torque to also delivered to the other wheel. The pitch of the helix determines the ratio.
Locking Diff: Not so clear on this one. Some can be manually locked, others automatically lock. When locked, the axles are solidly tied together (ratio is 1:1), and when unlocked, it's effectively an open diff. Usually used for drag cars, but some AWD systems have a locking diff in the middle.
(Just cause it's good to know

)
EDIT: I was not clear, a locking diff is
not a LSD. It's just a locking diff. And a Positraction is a type of locking diff.