you have mainly 3 types of LSDs
Hellical
Viscous ( vLSD )
Clutch type
Hellical LSD is a gear LSD
power is transfer to both wheels by the gears, one draw back of Helical LSDs is that both wheels have to has some resistance ( touching the ground ) for the LSD to distribute the power, if one wheel is in the air or on ice, the gears will spin the wheel with no toruqe twice as fast, this is the reason why hellical LSDs are also refer to as Torsen ( torque Sensitive ) RWD drivers overcome this by applying the eBrake when one wheel looses traction, the brakes will act as a resistance on the wheel with no traction, splitting the power back to 50 / 50. Hellical LSDs can be aggressive as with a sudden torque application to one wheel the other will grab instantly.
Viscous LSDs they are sealed units, that contain a silicone fluid in it, along with LSD couplin ( metal ), as one wheel spins at a different ratio in relation to the other, the different in speeds will generate friction and thus heat, once the silicone gets hot it will expenad moving the LSD couplin together and thus sending the power to both wheels to bring them to the same speed.
Viscous LSDs are maintance free, and the only thing to worry about when using them is not to get them overheated, overheating will destroy ( change the properties ) the silicone and the LSD will not work, so system will act as a regular open diff from that point and on. Most manufacturers prefer vLSD in FWD application because of the smoothness with which the power gets transfer.
Clutch type, name says it all, it has clutches inside it, thsi LSDs are very high maintance, under real racing they need to be rebuilt after each race or so, they can come in different ways like 1.5 and 2 way
this clutches are not intented to be installed on FWD cars, they are aggresive,
Note, I wrote this right before taking my lunch so I'm hungry and have not much time, there may be spelling errors or some information crossed out lol, but I'll revise it later or if you guys feel something is not right, corect me.