I've always thought Rick at ballparkmagic had a great idea for instant replay and having it so it doesn't affect the length of the game. (there is a lot of discussion and reasoning he puts forth, but I excerpted the basics below.
The replay ump watches a monitor constantly, most likely in one of the camera wells or in another place adjacent to the field designated for this purpose. He sees replays of everything, much like the viewers at home. He is concentrating on the game constantly just like all the other umps, only by watching the camera coverage instead of the direct action. He has access to every camera angle (home, visitor, Fox, TBS, scoreboard, etc.), though he probably doesn't need all of them. One good angle is usually enough.
After a controversial play, if there's an argument, the manager has the right to ask for a review of the play (politely, of course). The umpire on the field decides whether to allow the appeal. If not allowed, there's no recourse. If allowed, the replay ump's decision is final.
If the replay ump sees a potentially game-changing error (like the Mauer foul call or the non-double-play in Anaheim), he would have the responsibility to alert the crew chief immediately that the call was blown and needed to be fixed.