Possibly Spider gears

The only way to know for sure is to open it up. You might get an idea if you got all 4 wheels off the ground and then tried "driving" it to see if it still acts up with the weight off but that won't really tell you much as you can't make the differential work. Same thing if you hold one side and then turn the other, it might show you if it was a brake issue but not much else. It could be a bad gear or the pin may be coming free. Either way I would open it. If you clean the exterior and are careful you could re-use the fluid. I wouldn't wait as at the moment you may be able to save the parts, if they bind a few more times it could easily destroy the differential or break the housing or even the case and then you end up replacing it all.
Keep in mind that being it is a 4X4 you could still drive it some even if you find damage, you would remove the cross pin and those two spider gears, then put the cross pin back in to keep the axles in place and you could drive it with the front axle engaged. Not the best solution but it would at least let you have a vehicle until you could do a proper repair.
From the description I would suspect either broken or missing teeth on one of more. Unless it has some type of locker that is failing.

As long as it looks good you can re-use it, if it turns out that there is metal in it then I'd probably opt for new. Take some pictures and post them on here once you're into it. It could be binding in the differential as well if it has an LSD but until it's open you can't really tell.

LSD should spin the same direction, an open differential will spin in opposite directions if you turn one tire. The factory LSD in them is a clutch type, basically works using spring pressure and a couple parts that look like saw teeth. Those are the ramps that squeeze the clutches if one side starts to slip.