I'm going with the perspective of how did these games affect gaming in general. Some hold up better than others. Nonetheless, I'm looking at it from the perspective of which games showed ways to play games in new ways, defined genres, and otherwise broke new ground with players.
PlaneScape: Torment, SW: KotOR,BG II: Shadows of Amn, & Fallout: Sterling examples of the WRPG done right, showing the highly varied settings possible, deep engagement of player decisions into the outcomes, emotional impact of good open-ended writing, and how important a rich aesthetic is to the design of an RPG. Torment alone is the best narrative I have seen in any video game ever. Period. Bar none. I'd read that game like a novel.
Final Fantasy III/VI, VII, Chrono Trigger: Examples of the above for JRPG genre, with some ability to showcase how player decisions could impact even the traditionally linear genre.
TES: Arena-Morrowind, BattleSpire: Same, shown through open-ended generative play.
GTA: III: Created the entire genre.
Rome: Total War: Rome was the most approachable of the series. Total War married 4x and RTS, creating a different kind of depth than...
Command and Conquer, Starcraft, Warcraft: ...but these games also provide the cornerstones of many gaming experiences.
Halo: Love it or hate it, it doesn't matter that everything Halo did was done before. It was slick, approachable, easy to interface with, and surprisingly deep when you got into it. Halo single-handily put the Xbox on the map, made people want Xbox Live, and brought online FPS from the domain of LAN parties to everyone else.
CoD: MW: Again, love it or hate it, doesn't matter. It essentially created the genre which seems so dominant today.
Goldeneye: Did for FPS on console what Halo did for the Xbox.
Thief Series: Barring a few minor games, this really put the entire idea of stealth on the map.
Monkey Island Series, Myst, King's Quest series, Quest for Glory series: Some of the best in adventure games, a disappearing genre.
Silent Hill 2, Resident Evil, Amnesia: Dark Descent, Alone in the Dark, Resident Evil IV, and Eternal Darkness: Look, horror has gotten a bum rap lately. Even Dead Space has stopped being horror and become action with jump scares (which isn't necessarily bad...). Each of these games captures a different aspect of the genre. SH2 catapulted adult fears to the spotlight. RE and RE4 created whole new gameplay types for the genre, as did Amnesia. Alone in the Dark may have laid part of the groundwork for it as did Eternal Darkness.
Sid Meier's Alpha Centari and Civilizations II: All 4x games come from these. Alpha Centari never got the recognition of the Civ series because it's topic is a little unapproachable, but it managed to show 4x games could be highly thematic as well as offer a strategic depth very few games today could match. It holds up -now-.
Dragon Warrior/Quest: Do you know why there are JRPG fans in the US? Because way back when, Nintendo gave out copies of this game. They gave it away. The rest is history.
Ocarina of Time: Really, does this need an explanation? And while we're at it, the original LoZ brought for an open-endedness and lack of direction which set it apart from anything released. Now, devs are almost scared to allow a player to explore freely and try to spend time learning secrets. Someone always tells you what to do and how. Not so here. I kind of miss that.
Street Fighter II, Marvel vs. Capcom, and Mortal Kombat I & II: Defined and redefined genres.
Guitar Hero, Rock Band, DDR, Angry Birds: Bring on the hardcore gamer hate. These games defined new playstyles, brought gaming to communities and demographics which would otherwise never have considered it, and made mounds of money.
Mass Effect series: Do you know what no one really seems to note about this series? It's a high-concept project which blurs deep elements of two completely different gameplay styles. It required a big leap of faith to develop. It had to gamble on marrying two very different types of gameplay experiences and selling the players on an untested, unknown intellectual property. Then they had to world-build a huge setting. And they pulled it off. Mass Effect is one of the deepest gaming intellectual properties out there, maybe rivaling TES or Warcraft for world-building. (There are deeper properties, such as VtM Bloodlines, anything Warhammer, and so on, but most of these are imported from outside into gaming.) It was a crazy experiment and it showed devs that yes - if you do this well, the fans will come. And that matters if you want something other than repeated rehashings of the same old IP and concepts. And do you know what the fiasco about the ME 3 ending taught us? When they come, the players will still care about that IP just as much as fans of other media will.
VTM - Bloodlines: Such a flawed masterpiece. This is why execs should not mess with pushing the product out before it was ready. The haunted mansion sequence alone is worth a playthrough. Players who put up with the bugs got an action-RPG unlike anything put out in the past. See also: KotOR II.
A few other mentions:
Ico, Shadow of the Colossus, Adventure, Super Metroid, Super Mario World, Wing Commander series, Half-life, Doom, Quake, Unreal Tournament, Ultima Series, System Shock, Deus Ex, Maniac Mansion, Super Mario Kart Series, Grand Tourismo Series, Duke Nukem 3d, WoW and EVE, Zork, FTL, Dwarf Fortress, Eye of the Beholder, Missile Command, and many more.
