There isn't really a predefined objective. It's very sandboxy. Playing locally doesn't require much of an internet connection (it "phones home" when you start up, but that seems to be it.)
However, in the last couple versions they started adding little rpg elements. There's XP now (so you can level, but it doesn't affect anything yet.) There are (currently empty) towns, mines and a couple fortresses kicking around. I think I read something about them introducing NPCs in 1.9 (which can't be too terribly far away.)
Though now that Skyrim's out
