One popular game bug in the Quake series was "strafe jumping" [1]. It became a canonical movement technique that got ported into other games.

[1] http://en.wikipedia.org/wiki/Strafe-jumping

Another one is "wavedashing" in Smash Melee. Not really a bug, but certainly strange behavior that became vital in competitive play.

so many exploits in that game.. so many days i'll never get back.. anyone down for a quickie?

Similarly "skiing" in the Tribes series was a glitch in the first game that eventually became the later games' signature maneuver.

Snaking in F-Zero GX or Mario Kart. Especially in the former, it makes the game even crazier than it already is.

Just like the amazing amount of StarCraft glitches, ranging from making ground units fly to completely insine stuff like turning a unit into a trap that crashes the game for every player who looks at it (and after some time crashes the game for everybody). All by legal in-game actions. I can't find a link, but one comprehensive summary back in the old days suggested that at least some of those glitches may be intentional, given how improbable, weird and useful they were.

Anyway, an example list: http://www.entropyzero.org/Glitches.html.

I will never forget super-bouncing in Halo 2. Ever.