'Immersive' is one of those words that almost everyone has their own interpretation of.
For me, I'd say it's when a game world doesn't pause while you go fumbling through your inventory for half an hour. A game where what you carry is actually represented on your model. A game where, just like in real life, a knife to the face or a bullet to pretty much anywhere is going to do more than make you grunt and flinch momentarily.
And the sad thing is, almost every game that people have thus far listed do not match my interpretation AT ALL. The most glaring is Fallout/Elder Scrolls games - don't get me wrong, I love the games. I spent my Sunday playing Skyrim for about 5 hours. Playing through as a Breton mage now (just doing the College of Magic stuff, ignoring main story. Finished Dragonslayer story last night with my Nord axeman), and whilst blasting my way through another room of rogue mages I noticed my health was quite low - so I opened up my inventory. And proceeded to spend 5 minutes quaffing a few health and magicka potions, some resist magic and fire and frost philters, then I decided to double check that my activated spells were the best for the situation ... annnnnd ... all good. Back to the fight. BZZZT FLASH BANG they were dead and I had won.
Compare this to a moment of combat in Mount and Blade:
OH SWEET JESUS HE'S GOT AN AXE ... *wince ... parry!* PING! damn he's fast! *woosh thud* .. here he comes again!! *swoosh ping ping thud ping swoosh chop* HRRAAAA ... *Crumple* He's down .. right, *looks around quickly ... sees enemy cavalry making a beeline for him, lanced crouched and aimed* Oh shii ... *SKEWER!*
Now THAT is immersion. There's no inventory micromanagement in the middle of a battle. If you don't have the right thing in your hand to start with, you're gonna die. Quickly.
A lot of people in this thread seem to be just describing a game they like, and using 'immersion' like you would use the word 'good'. Seriously ... Final Fantasy? Are you fucking kidding? A game where bitter foes line up and take turns using their most powerful attacks on each other? Sounds like fucking Roshambo to me.