When I play games, the most I get out of them is a "oh, this is neat" thought but not much else. Seems like this thread is about the much else.
I must say that this generation has done a much better job at squeezing tears from the audience. The first time I ever felt anything was in Majora's Mask; I don't know what it was exactly, but it was just a sad game with a sad-empathetic antagonist. I felt that depression through the characters, even without spoken dialogue. So, if done right a game can express emotion without a strong plot.
Many say this about Bioshock. I didn't feel much besides "oh, this is neat." The world is wonderfully designed and has a great "oh man, I'm going to die" atmosphere. The plot is well thought out (rehashed), neat; while there's supposed to be a sense of betrayal, I just thought "oh, this is neat...those guys are clever." I played the game much like any other because it was very objective based; this got in the way of the moral choices that came with the little sisters because of the disconnect between mission objectives and little sisters. I played the game with that mindset, but then at the end there was the orphanage, here I saw the result of my actions. I felt regretful; the little sisters were scared of me. Had the game been more progressive with the end (orphanage scene), the whole experience would been what many claim it to be.
A game many have overlooked, Call of Cthulhu: Dark Corners of the Earth is similar to Bioshock. What's so great about it is the fear one has of it. Many horror/scary games "scare" you with gore, "BOOs!," and usually the undead, but Call of Cthulhu is subtle. It has gore and a couple of "BOOs!", but you're not scare of that, you're scared of something worse, you scared of being scared. At least I was; it was just so great being scared of the unknown. This may be a biased view to get you to play the game. I've jumped playing games but have never been truly scared, except in Dark Corners of the Earth.
Fable 2 was game were I felt an emotional tidbit, the dog. I cared for nothing, really; the game was very shallow with moral choice, like the original. However, the dog was nice and I felt the need to get him back when he died. It was a short-lived emotion though; I now play the game with hardly noticing it.
I know people may not agree with my next emotional moment, but: Gears of War 2's Carmine. Everything regarding this franchise's plot is stupid, but I liked the father-son like relationship between Marcus and the rookie, Anthony. It didn't feel forced like Dom's. Many of us have experienced something similar to this in real life (unless you're a homeless bastard); that's why I like it, it makes me reminisce. It was nice hearing "You did good." Who doesn't like compliments? It sucks when he dies because it's like YOU died; especially since he's survived so much already.
Half-Life 2: Episode 2's ending, of course! One grows attached to Alix much in the way one grows attached to Fable 2's dog; except Alix talks with a [sexy] female voice, so it's a more solid bond. While I like Half Life 2's characters and would miss any of them were they to die; Episode 2 is a great example of empathy (or sympathy if you'd actually lost someone to an alien). The voice work and visual direction really shine in that scene; it makes one feel like they are there. Surprise, sadness, and dash of anger all wrapped up within a short amount of time.
Lastly, FarCry 2, The game's plot is intriguing enough if you don't bore yourself with the excessive missions. I didn't expect much besides neat choices, blowing shit up, and exploring (which I love). Upon my adventurous exploring, I died, and this dude comes to my rescue. He didn't look very likable, but I was grateful. Eventually he became my best friend by the game's standards; I didn't much care for him. He was some old dude that didn't look to friendly but saved my ass regardless. Anyway, at one point I chose to do a mission with him; he was at another mission point helping and was badly injured. I clear out the area, go to him, and try to resuscitate him...over and over. He dies, and I see my character's hands comforting the lifeless body. I was surprisingly saddened; this man had saved MY life countless times (5), and I couldn't return the favor. Like I said, I had not grown too attached to him, but I valued him, just like a real person's life. I'd have to say that that is closest human emotion that has come from a game. I treated those NPC's like humans. I didn't like some of them visually but appreciated their help.