I'll throw in a recommendation for Planescape:Torment. It is a completely different kind of role playing game then most of the others mentioned (vastly different than Baldur's Gate for example). Most rpgs seem to focus on combat and gathering treasure and the story is given second place. Not so in Planescape:Torment. There the story is extremely immersive (a bit of a mystery in that you wake up in the morgue with no memory and quickly learn that you can not die so you are trying to learn who you are). Since who you are is an integral part of the story you also get to make many decisions on what type of person/character you are. So Planescape:Torment offers you some real meaty actual "role" playing decisions. The potential downside to the game is that combat plays a somewhat secondry role (their are several combats and I really enjoyed them because they were important combats in the story arc, but not nearly as much "hack and slash" as in say Baldur's gate).
Another game which offers you plenty of "role" playing decisions and is very unique in both story and genre is Geneforge from Spiderweb software. In that game you are a shaper with the ability to create your own army of mutant creations which do your fighting for you (it too is a mystery where you are shipwrecked on a forbidden island and have to figure out why the island is forbidden and who or what shipwrecked you there). Plus making your own little pet monsters and running around with them is just fun.

Another thing I really liked about the game is that it is morally ambigous in that who seems to be "good" is later revealed to be doing some pretty nasty things and that who seems "bad" later turn out to be almost justified, except... (i.e. what is ultimately "good" to your character is a truly interesting choice depending on what your character values and not something that is pretty obviously good like stopping the world from being destroyed or whatever).
Teal