Open differential = whereby one tire is NOT mechanically linked to the one on the other side. Each side is individually powered by the driveshaft. The tire with the LEAST traction will do a burnout and the OTHER tire will just sit there, looking pathetic
Locked differential = whereby both tires ARE mechanically linked together. The driveshaft turns the ring & pinion and in so doing, causes both tires to turn at the exact same time
Spool = a LOCKED DIFFERENTIAL, whereby the housing itself is nothing more than the two splined shafts for the axles and a mounting for the ring gear. A simple, solid unit, primarily for drag racing. Lightweight and durable. But does not allow for turning without scrubbing (dragging) the tires during cornering
Mini Spool = replaces the guts in a Open diff, with the guts of a Spool. Less durable than a full spool because of size, but with similar characteristics.
Limited slip = typically a set of clutches PARTIALLY cause both tires to turn. When one tire slips, the clutches develop friction, thereby causing the opposit side to begin to turn. The tighter the clutches are shimmed together, the more the traction to the opposit side
Detroit locker = a mechanically locking differential, that will allow it to dis-engage for cornering - but still ALWAYS sends power to the tire that is going the SLOWEST (typically the inside tire) - and when ENGAGED - send power to BOTH tires, just like a LOCKED DIFFERENTIAL.
ARB locker = an OPEN diffenential (just like most every stock application) that uses an AIR actuated solinoid to ENGAGE and send power to BOTH tires, just like a LOCKED DIFFERENTIAL
Welded (aka LINCOLN LOCKER - named after the Lincoln welders) is when you've welded the internals of an OPEN diffenential, creating a mechanically linked axles (tires) of full LOCKER
OX locker = similar to the ARB only uses a cable to operate instead of an air line (I think, I think, I think)
Detroit Truetrac = a pretty cool design of a Limited Slip, that uses hellically cut gears instead of clutches to give good limited slip traction. The advantage is no clutches to wear out.
Eaton E-locker = similar to the ARB only uses a electic solinoid to operate instead of an air line or a cable.
Detroit Electrac = harder to find unit, that uses a Truetac set of internals, but when you engage (E-locker) the electric solinoid, then it becomes a true LOCKER. This unit is both a LIMITED SLIP and a LOCKER.
and there's probably a few other I can't think of right now.