Exploration:
Wind Waker I love just exploring an ambient environment. Wind Waker had it's flaws (islands that are clones of each other), but that feeling of exploring this massive sea, never knowing what might come up, is amazing. The graphics really seal the deal, making it feel like a fairy tale. Never have I felt more like I was on an epic adventure.
Metroid Prime I played all of Prime with the scanning helmet on, just reading the lore. The music, the attention to detail, everything made you really feel like someone exploring an alien planet, so very alone.
I know I should probably play morrowind and oblivion if I like exploration, just haven't done it yet
Strategy:
Advance Wars (1+2) The simple gameplay belies the much deeper underlying strategy. This is turn based tactics distilled to its most vital and rewarding core
Civilization 4 The Sid Meier formula works. You can play them for fun, creating an nation after your own choosing (want to play a slave whipping military empire or a peaceful, technology based republic? Okay on both accounts). Or you can play it as a strategy game, trying to go for different strategies or military pushes at different eras.
Alpha Centauri Another Sid Meier game. It does a couple of things better than Civilization. First, the factions are all individualized. Playing the Believers or the University of Planet or the Gaians are all radically different experiences. From this comes a much better story as well. The other leaders you fight really have personalities, both from ingame interactions and the quotes from the datalinks. Also, there is the unit workshop, where you can design any unit you want for your purposes.
Fall From Heaven A mod of Civ 4, but it is so different that it is really its own game. It trades a historical for a dark fantasy feel. The apocalypse is imminent, and depending on how far it has advanced, blights and dread creatures from hell will sweep the land. There are magic users, powerful barbarians, epic weapons, hero units and devastating world spells that can be cast once a game. Every civilization and religion plays extremely differently. A great overall experience
Europa Universalis 3 This is soooo much better than Civilization at historical feel. The mechanics are very in-depth, but once you learn them, man is this game rewarding. You can select any country from 1399 to 1850 (and I mean ANY country), and pilot them however you want, setting your own goals and personal "victory conditions". You can try to be the Iroquois and survive against European colonization, westernizing as quickly as possible. You can be the Byzantines, trying to found a new and glorious Roman Empire. You can be Ireland, trying to assassinate every Bishop in England. In my current game, I finally united Japan after 50 years of war, and have now made a proclamation that I shall rule every island in the world (Great Britain will be the hardest one...). This game has incredible replay value, further enhanced by the active modding community.
Competitive:
Starcraft Brood War This game has probably the most developed metagame of any video game out there. It is the game that made the term "Esports" perhaps not a joke. It's really not the game itself that is so good (though it does provide a good game with a balance between all the races and a balance between the strategies), but the competitive scene. I have hopes for SC2, but it has a lot of catching up to do.
Super Smash Brothers 64 Yeah, I like this better than Melee. It is slower, and so allows more thinking versus twitch reflexes (which makes me much better at it). Simplicity and fun distilled. I will grant that for casual games, Brawl is probably the best in the series, but Original is the best for competition in my mind.
Adventure:
Shadow of the Colussus This could have probably gone into exploration too. I'm sure you've heard it all before: the beautifully crafted environments, the feeling of awe and insignificance as you stare up at the beasts, the sadness as you slay something innocent and primal. There is nothing here other than what was absolutely necessary, resulting in a game of minimalist perfection.
Portal Like any good puzzle game, Portal gives you a simple tool and then builds more and more complex situations based around it. What makes Portal stand out is the uniqueness of the tool. Using the portal gun is just fun, and forces you to think in a very different way. Additionally, the story was perfect. Just like SOTC, there was nothing here that wasn't absolutely necessary.