God, I think my favorite game of the decade is Cave Story.
At least as far as non-competitive games go. "Influential" is fuzzy stuff. But lets try and track this a bit. I'm going to use Kazz's list of possible games as a base, so hopefully I don't forget anything.
2000: Diablo II. Jesus christ how did they put anything else. I can't believe Gamefaqs put Chrono Cross, which it seems most people don't like or haven't played. Every year there are diablo II clones. I know a ton of people here got Torchlight. Guildwars is practically an 'mmo' (snrk) based on D2 and after 10 years, the Diablo brand has not diminished.
Deus Ex also seemed potentially influential, but at the same time no one needs help slapping RPG elements on every genre ever.
Also the sims. Your girlfriend is playing a videogame? GASP
2001: GTA3 influenced a lot. This just pushed the whole sandbox cancer we see in every game before it. All similar games before GTA3 were garbage. Halo is a tossup. On one hand, it's made a lot of money and spawned imitators, and on the other hand it's a goldeneye clone with aliens, an FPS on console (how unique!) and basically didn't do anything partially interesting. What it DID do though was somehow get really popular and make the xbox legitimate. I guess it also made every FPS ever on consoles to have netplay? Maybe? It might be less influential and more 'the right game at the right time'. Anyways.
2002: Again, Kazz is right. War III. And not even because of War III (like Kazz said too). Now we're getting all these DOTA clones released... Also every tower defense game ever.
Also Animal Crossing. Now your girlfriend plays two games.
2003: Prince of Persia. The start of all that free running BS. KOTOR came out and was good I guess, but it really didn't influence anything! A lot of fun games ever came out in 2003, but none of them particularly important.
2004: WoW. Not only did this destroy most of my friends, but now their girlfriends were back to only playing one game... but mostly out of neglect. :|
Next? Half Life 2. On one hand it's just another FPS (abeit a good one), but it also spawns a fresh mod community, gives us a new engine, legitimizes steam and is pretty much just solid.
2005: Guitar Hero. Girlfriend is back up to 2 games (but only when company is around). Now your parents play! Never successfully imitated (Rockband doesn't quite count since it's the same people, different company), it has had only mild influence on games, but had huge cultural impacts.
God of War is sorta a big deal. Thematic battles and ridiculous presentation mixed with solid, but classic gameplay. This has a huge cinematic influence on future games.
SoTC comes out, gamers are impressed, and no one else notices or cares.
Game "everyone wishes was influential". Also geometry wars comes out and gives 360 owners something to play until something decent comes out.
2006: Wii Sports. Lets all vomit in our mouths a bit. How many people own a Wii AND JUST THING? And they STILL PLAY IT. Also it has spawned ENDLESS SHOVELWARE. It has even made Nintendo make shovelware! SDFADF@#$@#@F@#
Some good games were also released that didn't influence anything ever.
2007: Peggle. Let everyone but your girlfriends vomit. Now they have a ton of games. You also play it too, secretly at night when no one can judge you.
TF2 is a bit too general. The Orange Box. The best deal in the history of gaming ever, that basically flat out said "PC gaming is not dead". Mass Effect comes out and manages not to influence anything despite being good (No one seems to try and be Bioware? Same seems to go with Besthesda, but thats a different story!). Call of Duty 4 did like, something. It was a big deal for a lot of reasons I cannot exactly pin down.
2008: Oh god this is getting to close to now. Wii Fit. Most second sellingest game ever, outside that thing that game with your second wiimote. Lets all just be disgusted. This is the second game your uncle owns, outside of wii sports.
2009 and 2010 are silly to do right now obviously.
