This page is a good one to bookmark:
http://www.therangerstation.com/tech_library/index.shtml
It's the Tech Section for TRS
Has an AXLE section with Locker info
An OPEN differential will send torque/power to the easiest axle to turn, so once a wheel/tire looses traction and spins it gets ALL THE POWER, lol, but they are the least expensive to build and install
Limited Slip uses equal rotation to divide the torque, if one axle starts to spin faster than the other the clutches or gears in the differential give the faster axle less torque, so opposite of OPEN.
Ford's Trac-Lok(and GM's Posi-traction) uses clutches, Torsen uses Gears, so a little better than clutches, which was why it was used on the Level II's
Locking differential mechanically locks both axles together, so they
must spin at the same speed, torque is equally divided to both axles.
This provides the best traction but also doesn't allow going around a corner because the inside wheel must turn at a slower speed than the outside wheel, so outside wheel will skip/drag to match inside wheels slower rotation, this will break something in high traction situations, i.e. dry pavement
So a Locking differential will have an ON/OFF mechanism, it can be mechanical, but most are either electric or vacuum operated from in the cab
They can't be "automatic" since that would defeat the purpose, Limited Slip are "automatic", so an automatic Locker would just be a Limited Slip, lol.
When off road, or low traction situation and traveling more or less in a straight line you would Lock the differential for best traction, small differences in wheel speed wouldn't matter too much because it's a low traction situation, so a wheel could skip/drag without breaking anything