Note these games may not be the FIRST to do it, but they are the games that revolutionized them and made them and are the ones that influenced gaming media. Here's what I've noticed so far.
Halo: Regenerating Health. (Nearly every game ever now)
Uncharted: Seemless Cinematic moments during gameplay,it's become more popular now.
Gear of War: The Cover System (Games have started using a cover system in more ways then one since then, including blind fire ect)
Grand Theft Auto III: Help set the impact of sandbox games, even now where games do their own style, the collectables, how side-missions are handled, mini-games, stealing vehicles ect ect are usually borrowed from the basic ideas of this game. (Note, games like Skyrim, Crysis, Far Cry, ect do NOT fall in this category)
Vanquish: The Sliding Mechanic. (Although games don't have it as INSANE, after Vanquish it is noticed that games have been adding a sliding mechanic when close to cover now days. This is usually seen as a Vanquish x GoW combination)
Resident Evil 4 & God of War (QTE) - Games use them now, sometimes way more then others, but RE 4 was the one that was mostly praised for using QTE's to create different scenarios. God of War incorporated them into more gameplay level, which has also been seen in games now days. So they both get their share of spotlight here.
Assassin's Creed: Free Running. - In some shape or form a good amount of games now use a type of free running Mechanic when they can, Infamous, Prototype, Sleeping Dogs ect are a few that do this.