Just a heads up about Morrowind: The Game of the Year Edition is riddled with bugs and stutters when playing on a 360. Your better off playing it on an original XBox, even though it is technically compatible with 360. I absolutely echo this suggestion, though.
I haven't seen Oblivion mentioned. That is my favorite entry in the Elder Scrolls series. While Skyrim was epic, it took some of the more nerdy RPG-stat stuff away, which I missed. Oblivion had a nice balance of that and real-time action.
I also haven't seen the MechAssault games for original XBox mentioned. Fan-freaking-tastic mech games that are heavier on the arcadey action than the boring simulator stuff the MechWarrior games gave us.
Jade Empire for original XBox is a fantastic action-RPG that is quite unique. It's set in Ancient China and uses a martial arts based battle system where you can learn different techniques to add to your magic attacks or special physical attacks. It also has a really cool good vs. bad system that really does affect the story.
Bionic Commando on XBox 360 is an overlooked gem. Such a fantastic action game that truly tests your skill. And the grappling mechanic, once you get the "swing" of it, is quite addicting.
I of course echo the suggestions of Skyrim, the Halos, and Star Wars Battlefront. Many hours spent in those games.
Panzer Dragoon Orta is another original XBox game I highly recommend, that has already been mentioned. When people say that games should be considered art, this is a game I think of. The graphics, the atmosphere, the setting, and the music all just combine into this wonderful, mesmorizing experience.
If you like Diablo-esque hack-n-slash RPGs, then you MUST play Baldur's Gate: Dark Alliance 1 & 2 and Dungeons & Dragons: Heroes. Highly addicting loot-fests!
If you like open-world RPGs, Kingdoms of Amalur: Reckoning is one that does not get the attention it deserves. Amazing, and beautiful game with names like R.A. Salvatore and Todd McFarlane (of Spawn & Venom) working on the design. It feels a lot like Oblivion or Skyrim, but with a more satisfying battle-system.
I could keep going and going. I've been an XBox guy since the first system (though this generation, I feel I might have made the wrong choice).