Shocks with resistance on both the compression and rebound strokes are called what?
Shocks with resistance on compression, but rebound automatically are called what?
What is a 'monotube' shock?
Of the first two referenced, what is preferred offroad and why?
What you described first are hydro/hydraulic shocks, like the Doetsch Tech DT3000s, Rancho RS5000s, and other shocks like that. These are usually your lower-end, budget shocks.
Shocks that rebound automatically (they still provide resistance though) are called gas shocks, or sometimes "nitro" shocks, or gas-charged shocks, etc. These are going to be your Bilsteins, nearly all monotube shocks (and many twintubes), Old Man Emu shocks, etc. I believe that all factory Jeep shocks are gas shocks as well (every one I've seen has been anyway).
As far as which one is preferred, that just comes down to, well, personal preference, haha. Me personally, I prefer high pressure gas shocks. And I think most folks do too because nitro shocks are going to feel firmer than most hydro shocks, and they'll maintain control better because they won't aerate and cavitate like a hydro shock will during rapid cycling (ie: fast bumps, etc).
The first link that muddeprived posted has some good info on the technical differences.
FWIW, I have shocks on my truck that have no rebound (downward) dampning, and it sucks!! I have an 07 Nissan Titan, and the front wheels will drop right off of something and go all the way to the bottom with no dampning of the wheel at all. I see no off-road application for a shock that doesn't control compression and rebound. HTH!!
Sounds like you've got busted shocks then... I can't imagine any shocks having no rebound dampening.