Let's see here.
MMO - Classic EverQuest. I just cannot get back that feeling of exploration and adventure before everything became instanced and all "secrets" were on spoiler sites. Alot of people take it for granted that within days of a new content release the players will now know everything there is to know about it, but "back in the day" we used to have fucking LEGENDS about stuff that we would whisper in hushed tones while camping Lower Guk and avoiding endless trains caused by bad monk pulling. I like WoW and everything, don't get me wrong, but there was no mystery in it and no sense of wonder for me. Maybe it's like how you can never forget your first love, I don't know.
ARPG - Diablo 2. So much fun, so many builds. So much god damn replayability. Diablo 3 is shit and I played it for 12 hours before never looking at it again. Hearthstone is a better game than Diablo 3 and that's kind of sad.
4X - Alpha Centauri. Unique Factions, Crazy Tech Tree, Interesting Events and Unit Design. Love, love, love this game. Better than Civ IV even and that's a close one. Honorable Mention: Master of Orion 2 and Galactic Civilizations 2. Space 4x is a small, but precious genre.
RPG - Baldurs Gate 2: Shadows of Amn. The best CRPG ever made, hands down. There are lots of RPGs and JRPG's that I love for various reasons, but BG 2 has everything that those other games does and it has it in spades. I laughed, I cried, I shouted bloody revenge.
RTS - Star Craft. Blizzard again here. Why is this game so fucking good? WHY??? It is easily one of the best designed RTS's I've ever seen. Easy to learn, hard to master, well balanced with lots of viable strategies. I guess there is a reason why so many people consider it basically the defining game of the genre, or the yardstick by which all other RTS's are measured against.
FPS - No One Lives Forever. I am not into "hard core tactical" FPS's or ultra realism. I was a huge fan of Duke Nukem 3D and played that game for years until I found NOLF and fell in love with Cate Archer. I'd say this is probably the first story based FPS I ever played and I can't think of a single one since then that's had me both simultaneously in stitches and on the edge of my seat. The fact that this game is now stuck in License Limbo because no one can figure out who owns it is fucking maddening. As far as I'm concerned Cate Archer needs to come back and show the "new" Laura Croft how to kick ass in go-go boots and a totally mod catsuit.
Stealth - Thief 2: The Metal Age. Part of the reason why I love this game is because the aesthetic is entirely out of Fritz Leibers "Fahfrd and the Gray Mouser". That sort of dystopian grim-dark fantasy world of low adventure and intrigue is just something that you can't find in most video games these days. The fact that not only did this game draw upon these incredibly inspirations, but also delivered amazing tension and action sequences really had me hooked. I consider this probably the finest stealth game ever made.
I guess most of the games on this list are from either the late 90's or early 2000's which sort of makes sense considering I'm almost 40. Also I wouldn't say that there haven't been games I've played since these that I've really loved, but these were my "firsts" that just blew my fucking mind or left a lasting impression on me that in some cases has endured for decades.
There are some hard calls on that list too, for instance I could go on and on in the RPG category about how much I loved Planescape or Fallout, or even older titles like The Bards Tale (god did I spend so many hours playing that game), so this wasn't easy to put together.