As psychedelics like LSD are reentering more scientific labs for mental health research half a century after conservative governments clamped down on their recreational misuse, the LSD we're interested with in motoring is the limited-slip differential.
A differential is essentially a mechanism in the powertrain that connects one shaft (coming from the source of power, ie. the engine) with two more shafts (leading to the wheels if it's a front or rear diff, or to the front and rear diffs if it's a centre differential).
An open differential has what is essentially unlimited slippage, where one of the two, let's call them output shafts, can be driven and take all the power from the input shaft, while the other output shaft can be held stationary. In a two-wheel-drive situation this makes it easy to drive, because you're more likely to just spin the power away on the partially-unloaded inside wheel, but that also wastes power that could have been used to accelerate the vehicle on a track, or to get it unstuck when off-roading.
As the name suggests, an LSD limits the difference between the two shafts that are driving that pair of wheels (or driving the front and rear diffs in the case of a centre diff). There are multiple designs of LSD to achieve this, and some are more complicated than others.
There's also the brutally simple locked differential to completely remove any difference at all. Some older 4WDs can also have one or more diffs locked by a lever or other simple mechanism before setting off on tough terrain. Plus, there's a self-locking design in some old 4WDs and in some old muscle cars (the 'ol Detroit locker, and similar) which has a mechanism that engages when a wheel starts slipping, but disengages to be open when coasting.
Each design of LSD - and each combination of their use for an all-wheel-drive vehicle with one, two or three LSDs - will result in different capability on or off road, as well as different behaviour to brake, throttle and steering input.
You may recall that understeer is when the front doesn't turn as much as you asked it to because the front tyres lack enough grip, and oversteer is when the car turns too much because the rears begin to slide.
The choice of differential (or differentials) has an influence on the effect of your control inputs as you manage, or more typically try to avoid, either of those situations.
In a RWD - and no road car should have one, it must be said - a locked diff is great for straight lines but quite a compromise anywhere in sight of a corner. They need pretty clever suspension setups (and/or lots of front downforce) to avoid excessive understeer, especially in slower corners. They're also far more prone to lift-off oversteer, which can make them very difficult to drive in low-grip situations like rain.
The self-locking design doesn't like engaging quickly so it's only suitable for slow situations, making an LSD the better solution, but the reason each design of LSD behaves a little differently relates to how it actually distributes the power across the two output shafts, and how much torque must be transferred to the one turning at a lower rotational speed.
If you've had a play around with differential settings in some sim racing titles you'll probably have wondered what the "power", "coast", and "preload" settings do (and some sims explain this, or maybe give them different names). Power denotes how much torque will be transferred during acceleration, and coast how much is transferred under deceleration. Preload determines the minimum torque between the two output shafts (so that it's never fully open).
In most cases these are adjusted during assembly. Too tight for power or for coast and you get the same downsides as the locked diff. So for competition, it's a question of finding the right settings for that track.
However, there are some really clever electronically-controlled differentials which adjust the amount of slip in one or more differentials either at the twist of a dial, or at the behest of a computer (which will have sensors everywhere, and other systems under its command like stability control), to further manipulate and help control the amount of oversteer and understeer.