Here is what immersion is for me. At this point I don't think that anything I write here will be a major spoiler for anybody, but if you haven't played Mass Effect or Mass Effect 2, I'll warn you up front that I'm going to be citing from those games.
For me, this is what it's all about. Running into Ashley on New Horizon and being so damned happy to see her, and then getting completely chewed out, and practically spit on by her for siding with what she considered the enemy before I could barely get two words out. I felt terrible after that, not felt bad for my Shepard, but felt bad for myself. Seriously, I went back up to the Normandy, and roamed the ship for a time, and then just stood in the airlock because I didn't know what to do. For me, it wasn't like she'd gotten angry at the character I was controlling. I'd invested so much time in that universe, and in those characters, that it felt like she was talking directly to me. To top it off, when I finally ambled my way up to the personal cabin, I learned right then and there that your fish die if you don't feed them. When I looked at the lifeless floating fish in the tank I practically threw my hands up in the air.
For me, that's immersion. When a game can trigger, in you in real life, the feelings and emotions it's triggering in the characters.