Way too many examples coming to mind. Just going by favorites:
Batman: Arkham Asylum - isn't exactly the first third person beat em up, but the combat's auto counter was revolutionary and influenced many games that came after it (Sleeping Dogs and Deadpool, for instance)
Quake 3 Arena (the PC version) - I don't think it's the first arena type shooter, but it's the one that became a staple and was used more in e-sports. Also, it became the de facto benchmark for hardware. It was the Crysis of its time but unlike Crysis, Q3 actually had a fun game underneath instead of just a $60 measuring tool for e-peen.
X-Men vs. Street Fighter (Arcade, sega saturn, PS1) - X-Men: Children of the Atom was the first game that had the double jumps and screen filling supers that defined the VS. series, but XMvsSF was the first that pitted two licenses from different companies. Now we have Tatsunoko vs. Capcom, Street Fighter vs. Tekken, SNK vs. Capcom, Battle Stadium DON, etc.
Resident Evil 4 (Ps2, Wii, Gamecube, PC) - was the first in the series to improve the tank controls, made the survival horror genre accessible to a wider aucience.
Mario Party on the N64 - when it came out, rival companies started coming out with their own first party boardgame/minigame titles (Sonic Shuffle, that weird thing with Crash Bandicoot)
Devil May Cry (PS2) - is probably the precursor to the hundreds of god of war-type games. Puzzles, platforming, combo-reliant combat. An annoying "super cool" main character, and QTEs. The first one is the best example because you can still see that it was originally meant to be a Resident Evil game.
