First, Moose said the stoegers are good, but will beat you up in the long run? What exactly is meant by this? According to Sheridan it sounds like a pretty good gun.
I think he's referring to a good amount of recoil coming from that gun.
Secondly, Big Uncle said a pump shotgun with different choke tubes would work fine, I'm new to this and have no clue what that is, so how would I make sure I have different choke tubes?
Choke tubes are not just for pumps. My double barrel side by side has removable chokes. Chokes are tubes you screw into the muzzle of your barrel that constrict the diameter of the muzzle to varrying measurments. The smaller the muzzle the tighter your group, the wider the muzzle the more open your group.
I found the Remington 870 that Vapodog recommended on gander mountain's website
That is a 20 ga shotgun, does it make much of a difference?
The 870 comes in 20 gauge and 12 gauge (and maybe 16 and 28 too, not sure though).
Last, if possible I wanted a shotgun that I could eventually hunt deer with if my brother gets back into it. I'm not sure if this is possible in my price range, and if not then so be it. But if it is possible, what would be the most befitting type of shotgun for me?
Most any modern shotgun will shoot slugs just fine. Slugs are what you want when hunting deer. You can get a shotgun that has a rifled barrel, get a gun that has a smooth barrel but comes with an extra barrel that's rifled, get a gun with a smooth barrel and rifled choke or just use a smooth bore shotgun and rifled slugs. All work fine.