Let's see...
Maniac Mansion -- steaming stamps off the envelope. I was so shocked you could actually do that when I attempted it on a whim.
Privateer -- the first time space pirates hail you like a friend if you've spent enough time shooting down cargo vessels. The concept of enemies in a game becoming your allies if you played differently was completely new and foreign.
Final Fantasy 8 -- the dance/waltz segment. That was the very first time I was blown away by CG in a game. It reminds me of how far we've come. CG has become passé and only real-time graphics seem to impress anymore. (On a tangent, everyone always talks about FF7, but I think FF8 is underrated. It's really one of the only mainstream games that I can remember that had the guts to attempt to tackle love as a theme. It didn't really succeed with that, but no game before or since has either. With so many movies that explore love, why so few games? Even as a side-story, it just isn't there in any meaningful way. Indigo Prophecy sort of tried to go there, but I can't really think of any others.)
Street Fighter II -- playing this for the first time at a friend's house and thinking that console games could eventually be just as good as arcade games.
Super Mario Brothers -- isn't first playing this on most people's lists? My grandma bought me an NES, which was the first game system of any kind that I had ever played. Whenever Mario would jump, I would whip the controller up and around as I pressed the buttons.
Resident Evil -- when the zombie dog jumped through the window in that first hallway and I began to realize that a game could actually be scary.
Morrowind -- that first realization that I could leave that little town in the swamp whenever I wanted to. And that I could kill the shopkeeper and take his stuff before I did.
Gears of War -- the first 'next-gen' experience that really blew me away. It didn't really hit me until I got out of the prison and my HD-TV purchase suddenly seemed like a perfectly reasonable investment.
Doom -- playing this on my uncle's computer and experiencing that first-person view for the first time.