A differential is a device that allows the tires to turn at different rates. When your truck goes around a corner, the tires turn at different rates.
Experiment: take a pencil, and roll it across the desk. Now, while holding the pencil, attempt to make it turn. Notice how the inside of the pencil wants to turn slower than the outside? [Better experiment: find some kids toy that has wheels on a solid axle. Not a Matchbox car, but maybe a sandbox toy. This would show even better how the outside wheels need to turn more than the inside wheels.]
In 2WD, at least in most modern RWD biased setups, the left and right front wheels are "disconnected" and the front differential is along for the ride. Otherwise front and rear are alike: as you turn, the differentials (or "diffs") allow the tires to turn at different speeds as necessary. Older 4x4's used manually locking hubs on the front axle; unlocking those unlocked the front axle. Leaving them locked but the transfer case in 2Hi meant the front diff was allowing the front tires to turn as necessary, but no power from the engine would go to the front axle.
Newer setups have automatic hubs of various sorts to disconnect the front differential/driveshaft (sometimes none at all); one rub of understanding 4WD is to realize... there are a multitude of systems, and each is different. [Including front biased 4WD setups, like Subaru and Honda.] Most of what I am outline is for traditional part time 4WD setups, like yours.
The transfer case is an item after the transmission. It is used to split power between front axle and rear axle. "Split" is done in a few different ways. Old school part time 4x4 systems, the only "splitting" that occurs is that the front driveshaft and rear driveshaft turn at the same rate. Power is not split; they are forced to run at the same speed.
Here is where it gets interesting: due to the tires angling on their axle (in order to make a turn), the front wheels and the rear wheels turn at different rates when turning. In an old school transfer case one will notice binding when making sharp turns while in 4WD--the tires would start to fight each other. The front and rear driveshafts need to have slightly different speeds. An old way of doing that was to put... another differential into the transfer case. This would allow different drive shaft speeds.
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At this point we can talk about "open" differentials versus limited slip differentials. An open differential can allow one wheel to receive all the "power" from the driveshaft, while leaving the other with none. "One wheel peel" is one term to describe it; stuck is another. Open differentials are notorious for letting vehicles get stuck. But they are cheap (of course) and allow for smooth operation on good roads.
[A differential splits torque evenly between the wheels. It's kinda hard to visualize; but truth is, each wheel is receiving the same torque. The way the diff is splitting torque requires it to work against the non-spinning wheel: if you were to lift it up, it'd suddenly spin. You have to visualize the spider gears inside the diff to see why the stuck wheel gets the same torque, hard to explain. A differential does not split power nor shaft speed, but it does split torque.]
A limited slip differential has guts inside it to... limit wheelspin, or wheel spin difference. It has to be able to allow some difference, but not "lots". GM used a locker, option code G80, which would lock after one (or two?) wheel rotations, locking the tires together; but allowing normal turns in the unlocked case. Other kinds exist (positraction or "posi", among others), using different methods to limit wheelspin. A new method today attempts to use the ABS system to limit wheelspin by sensing a spinning tire, and applying the brake to just that spinning tire.
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So, a transfer case could be upgraded to have a differential too. But at this point you can see how having three open diffs might lead to... one wheel drive (of sorts). So a slightly improved transfer case could have 4hi locked and 4hi unlocked options (in addition to 2Hi, 4Hi, 4Lo and neutral), by using an open differential that could be locked when needed.
I do not know much about the "auto" 4WD systems on newer trucks, if they are using a limited slip differential, a locker or other means (viscous couplings are used on some setups); someone else can explain what it is you have.
