Cod4 (rapid increase in Live users and the way online fps functions.)
Halo CE (twinsticks)
Halo 2 (Live stage 1)
Mario 64 (first 3D game of substance and analogue stick)
Pong (first, truly succesful, home videogame)
No particular order, I'm sure loads more could be added. GTA3, Doom, Gran Turismo, Space Invaders, Skylanders, World of Warcraft, Quake 3 etc
Couple of reasons, kicked off the current day shooter scene, but also for it's perk unlock nonsense. Whether you like this or not is another thing, but without COD 4 emblazing this into popular conciousness,n it's quite possible Halo would still be the gold standard for how we do shooters.
Grand Theft Auto III
It's surprising just how well this holds up nearly a decade and a half later. The open world sandbox has been copied a million times, and rarely bettered. The level of play outside the main missions was in some cases a bigger draw, and inspired numerous other games.
Doom
Yeah yeah, Wolfenstein came first, but who actually cared? The term was Doom clone, not Wolfenstein clone. Plus, Wolfenstein wasn't designed to be modded, people hacked it to do it, Doom had that built in. The shareware version meant that I don't know a single person who had a PC back then that didn't also have Doom. It's success paved the way for basically every first person shooter ever.
Half Life 2
Not for the reasons you might think. It's requirement to install Steam helped firmly establish the service, and without that, we might have a very different PC landscape.
Bejeweled
I've no idea if this was the first one. I'm going to say probably not. But again, that I don't know what it was based on tells you a lot. One thing is for sure, no Bejeweled, no Candy Crush. Not only are they mechanically very similar, Bejeweled created the market that Candy Crush, and all those match 3 jewel games on the DS were flogged to.
Halo 2 would pip CE for me. The first was incredible, but unless my memory is failing me (again) H2 laid the foundations for tonnes of console lobby/matchmaking online stuff, and the net code was reliable.