Super Smash bros.
Completely changed the formula for fighting games. In Nintendo fashion, made the game welcoming for anyone through easy controls but with a skill gap for more dedicated players. It also made the game satisfactory by letting us knock out our enemies of the stage. Items enables the outcome of the game be unpredictable giving the players with less skill a chance to win against more skilled players. All in all it does everything right and everyone can enjoy it.

Mario, without a doubt.

Pacman and Mario.

Doom.

Wolfenstein came first but Doom made the FPS genre explode. Arguably the most important PC game ever created.

How could I forget Gran Turismo? Huge turning point for the popularity and growth of racing sims.

Legacy of kain series
Mass effect series
Sonic the hedgehog

Can I ask how? Sonic maybe but mario overshadows him for sure.

The only games I can think of other than mario would have to be Final Fantasy, WOW and maybe Gran Turismo. The rest of the really popular franchises haven't realy been around long enough for us to see a "legacy", or they just don't sell enough.

Gran Turismo - sim racing
Zelda - the model action adventure
Super Mario -platformer (Metroid and Mega Man also have lasting legacy on design)
Metal Gear or Thief - stealth
Doom - FPS
Final Fantasy - JRPG
Ultima - Western/Computer RPGs
Mario Kart - Mario spinoffs, initiator of kart racers
Street Fighter - most influence in 1 v 1 fighting games
Myst - immersive environment puzzles
World of Warcraft/Runescape - popularizing and standardizing MMO space, Runescape was free to play
Farmville - social f2p boom

DDR - rhythm and dance genre has used the same interface and concept in most games, also influential as a party game

SimCity - I think we have this to thank for pretty much all sim games and future strategy games like Civ

Doom, Wolfenstein, Unreal tournament, Half-life. Those games made and defined FPS and good online FPS multiplayer. Everything today especially Halo and COD are imitations of those games and wouldn't exist without them. Then there are franchises like Elder-Scrolls and lets not forget D&D which are pretty much responsible for great game modding and well polished RPGs in general. Without D&D and other P&P greats I doubt any of the game developers would exist today.