I entered this thread, saw Mass Effect and Skyrim put down for "greatest RPG of all time", and violently convulsed.
Disclaimer: My bunch includes games I haven't played that I think deserve the title just because of their influence on their genre.
FPS: Half-Life. Doesn't really wow anyone today because everything it pioneered has been copied to the point where it's become cliche, but this is where all those things started and originally worked so well. Little things like having you get items from supply crates rather than having them lie around randomly on the floor (as seen in prior shooters), and telling its story completely without cutscenes. For multiplayer shooters, Team Fortress 2.
RPG: Planescape: Torment. There isn't a better exemplification of what an RPG should be. Fantastic story, layered characters, engrossing world, and REAL choice and consequences---not those little one-of-three-moods things, and not "Do you want to accept this quest now or do you want to accept it later?" Along with the original Fallout, this game represents the pinnacle of the undiluted genre.
MMORPG: Runescape, because it's the only one that doesn't feel designed to leech your life and money away.
Strategy: Fire Emblem, because it's the only one I've ever really gotten into.
Horror: Probably Silent Hill 2, for reasons people have explained countless times.
Music: Rock Band.
Sports: WWE Day Of Reckoning 2.
Puzzle: My favourite is Portal.
Simulation: Skyrim, world's greatest hiking sim.
Action: Viewtiful Joe by miles and miles.
Fighter: Mortal Kombat (2009).
Adventure: Beneath A Steel Sky.
Best Story: Planescape: Torment
Best Franchise: Fallout, because 1, 2, and New Vegas are phenomenal RPGs and some of the best games ever made; Tactics is great for what it is, FO3 at least has pockets of atmosphere and some cool landscapes to explore, and Brotherhood Of Steel doesn't exist.
Other (Can't define genre): System Shock 2. It's definitely an RPG, definitely survival horror, and definitely an FPS, to the point where it really is something else altogether.
Also, did you leave out Platformer on purpose, OP? In either case:
Platformer: Rayman Origins