(not mentioning exact games since sequels are the same, more or less)

StarCraft in RTS and eSports
WoW in MMORPGs (every WoW-killer has been more or less bad)
Diablo (it has had quite a few competitors in recent games, but I still think that it defines the hack-and-slash action RPG genre)
Counter Strike for multiplayer FPS (though I guess it has been losing recently to CoD)
Madden NFL (not sure that I know of any other American Girlball games)
Sims (has been dominating the genre since forever)
TES for singleplayer open ended RPG

For me personally no open-world RPG has bested Morrowind yet.

For corridor shooting I don't think anything has bested FEAR yet.

I think COD has taken the multiplayer FPS crown a long time ago.

DOOM. Popularized modding and FPS games, retains excellent gameplay up to this day.

Super Mario Bros 3. Ever since then, platform games have yet to surpass it.

I would give that honor to Super Mario World, but at least our choices aren't that far off.

Chaos Strikes Back for real time "blobbers" (Legend of Grimrock type of games).

Thief.

Probably. Wouldn't know, kind of gave up on competitive FPS play after playing CS 1.6 12 hours a day during high school.

DOTT - point and click adventure games (no, Grim Fandango is not a point and click adventure...)

SCUMM engine made the definite model at how adventure games should be played. Use/talk to and the rest are still in use wih only minor tweaks in the interface. Even later LA games that made heavy use of 3D tech had the same basic principle.
Some would argue that basic commands were in use in adventure games since forever, but SCUMM was the latest breakthrough that defined the genre.
Therefore: Monkey Island

Chess?

Who let the intellectual in here!? :P

Oh and my 2 cents:
Hungry Hungry Hippos

Chess the king of board games.

...not match for The Unmatched Divine God-Emperor of Boardgames, "Go", but pretty good nonthe less ;)

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Sticking to boardgames - Blood Bowl of football like games as far as i am aware.