I would have to say Pac-Man and Street Fighter 2. Pac-Man was the first game I've ever played, it got me into videogames. SF2 is evil.

Hard to say, between Zelda: OoT and Golden Sun and Metal Gear Solid. I even beat Zelda 4 times just to fight Ganondorf again, until I realise I could save before the battle starts. Golden Sun is magnificent, the puzzles were great and I like the characters a lot, the summons, just brilliant. MGS the characters were great, the story too, the two endings(one is driving Otacon and one is driving Mery), the psycomantys battle, the torture room, sniper wolf duel, do I need to say more?.

The first game to ever get me emotional was Link's Awakening.

*Spoilers*

When you finally defeat the last nightmare and save the Windfish, he tells you it was all just in his dreams (if you already didn't figure that out around level 6). Then as "Ballade of the Windfish" plays with all the instruments, pieces of Koholint Island start disappearing, along with all your friends unknowingly just fading out of existence, including Marin...

The one other game that got me was Ocarina of Time. There were two parts. One was the epic-ness of the final battle which gave me goosebumps and shivers down my spine.

The other part was the credits in general. First was when Link puts the Master Sword back in the pedestal, and Navi then flies away. I actually voiced the words "Where's Navi going?". Then there was going from scene to scene, seeing every one celebrating or just looking at the scenery of the places you've been. Finally, was when Link goes back to visit Zelda as a child and then they freeze frame and make the picture like a photograph.

I'm going to give a very honourable mention to the last fight of Wind Waker. That was just amazing. I actually half wanted Ganondorf to win.

*End of Spoilers*

Hmm... It seems as though this is why I'm attached to the Zelda series so much. They're the only games that seem to arouse my emotions.

Zelda Ocarina of Time and Majoras' Mask

This question will make my answer significantly more RPG-centric than a "What games are your favorites, etc." type question. (not in order)

Ogre Battle 64
Planescape: Torment ("What can change a man?")

Final Fantasy VI/III

Don't mean to start a long debate, but I really disliked Final Fantasy X, and I can't tell if I might've at least moderately enjoyed it without the voice acting. The fact that I felt like I could do a more convincing job than 60% of the voice actors made it hard to suspend my disbelief and get at all involved in the story.

The sphere grid was fun, though.

Also, Link to the Past, to a certain degree. I can still remember enjoying playing that so much when I rented it at age... 6? 7? that I made my parents buy it for me. It's still one of the most perfect and complete games I've ever played, in much the same way that Final Fantasy III/VI is.

FF X and Mario 64.

FF X- Still my favorite RPG, I dunno it just clicked so perfectly at the time and nothing has clicked as well since. Sure it had its flaws just like everything else in life, but I still consider it the best game I've played. First game I cared about the characters enough to say that it actually effected me personally. Took a really cliche story repeated many times (saving the world, the girl, the guy and whatnot) and perfected it.

Mario 64: The first real 3D game I played, the first game that gave a player an open world. So many great games since have also had open worlds but this was the first for me and so it left a huge impression. Everything from GTA to Shadow of the Colossus now owes the realistic 3D open world in console games to this game.

Shadow of the Colossus, Ico and Okami get honorable mentions for being both art and game and so they were all amazing experiences.

GTA 3 for being the first game that really directed the player away from the usual ideas of linear questing and objectives by having a world that can be played endlessly with no objectives and still be good.

I know

I didn't want it to be what's your favorite game.  A game can be really meaningful to you without being your favorite, and of course RPGs will be the most common answers because they are more character and plot driven than most games.  

Final Fantasy Tactics (PS1)

and

Ogre Battle 64 (N64)

and

Earthbound (SNES).