If we try and sort out nostalgia of specific situations (for example, my large dorm counted as one giant LAN network, so even before xbox live and stuff was super common, there were lots of games of Halo on the LAN all the time, and you were playing with the same large pool of people, so there were fierce rivalries between floors and shit... but that could have just as easily been a different game instead of halo.)...
I have to mention super smash bros 64 even though it's before 2000... Especially for games that don't rely on plot, it might hold up better over time than anything else, it's still fun as shit today. This was probably my favorite multiplayer game ever to play against friends. Also shout out to Mario 64 (only an OK game but the first truly legitimately 3d game I remember, it felt like such a huge step forward) and Occarina of Time (amazing music, good environments, felt like a big adventure. And like Mario 64, playing it really felt like such a big step forward).
Post 2000:
GTA 3: the plot was pretty shitty, and the out of car gameplay was mediocre, but the open world driving around the city sandbox was really awesome at the time. One of the weaker games on the list, but like Mario 64 it needs recognition just for feeling like such a step forward.
*We can also give a bit of similar credit to Assassins Creed. It was a very mediocre game in many ways, but it was the next big step forward as far as I remember in terms of "hanging out in a big 3d city." Not a good enough game to show up on the list by itself though.
NCAA 2004: Probably the best version of the game, and extremely fun as shit. Also the first year with online multiplayer. Sadly 05 sucked ass, and while it recovered a bit, it never seemed as good. With only a few improvements / features and a graphics update, would actually hold up pretty well. This might be my favorite 1v1 multiplayer game of all time. If they made a remastered version of this game and lots of people played it online, I would probably quit my job, stop showering, and just do that all day.
Socom 2: Hit detection was shit, cheesing the third person camera around corners played a major role, and there are a number of ways the game and matchmaking could have been way better, but the teammwork and mic stuff was awesome, and the lack of respawn made it so much more intense and exciting.
Mass Effect 1: This game might be my favorite single player game of all time. Of course the inventory system was shit even at the time, and the combat has improved a fair amount, but I think it was the best mass effect game from a plot / role playing point of view. The environments were so immersive, and I felt like I actually knew the characters. Ashley, Garrus, and Tali especially felt almost like actual friends from real life (Liara comes on strong in the later games, but I thought she was just OK in the first one). It didn't have a great soundtrack like Zelda or Star Wars that you would listen to outside the game, but it did pretty well in the moment. It was probably he most I've ever been immersed in a game.
It was also the first real "next gen" 360 / PS3 game I played, and like Mario 64 and GTA 3, it marked a major technological stepping stone in my gaming career. Game fall into the "before Mass Effect" and "after Mass Effect" categories from an expectations point of view.
Compared to Knights of the Old Republic (which was also a good game with an amazing plot), I really loved how we kept the same roleplaying / dialog options / interactive plot experience, but added in combat that you actually controlled. So many role playing games were just boring mathematical combat where you just click on shit and dice rolled. Mass Effects combination of role playing and actually fun combat was amazing. Also I loved having your character voiced, this was a major step backward for me when Dragon Age came out.
Sadly, then EA took over, and while the combat and graphics got better, the plot and roleplaying and shit went downhill in the sequels.
NHL 12: I could credit a couple different NHL games, but 12 was the best. I played these games almost exclusively for one reason... in 2009, they added a mode called EASHL where you only controlled one player, and played on a team of 5 humans against a team of 5 humans, just like real life. My friends and I made a team and played the shit out of this game like 5 nights a week. In the modern era of "cooperative multiplayer," where you often play WITH friends against random strangers online, this was probably my favorite game.
Modern Warfare 2: I don't generally like fish bowl arena respawn shooters, and I think the CoD series has gone way downhill but I played the hell out of this game. And for all the things I don't like (like the small team respawn, or the presence of killcams) the fundamental gameplay of this game was great. Boiled down to the simplest level of "me and this dude are trying to shoot each other," this was probably my favorite shooter game. This game is also maybe the only game I've played where (through a combination of gameplay and just as importantly map design) sniper rifles felt fun and balanced. Sniper rifles were worse at short range and better at long range, yet they weren't totally garbage except at very short range. It was also rare to get killed by a sniper rifle from like a mile away where you couldn't even try and respond with an assault rifle like in Battlefield or something. But it also wasn't like later Call of Duty games, where sniper rifles were so shitty (both through changes to the guns and a major shift in map design philosophy) it's not even clear why they are included at all.
To be fair, part of why I liked it was that the One Man Army perk was INSANELY overpowered, and yet for some reason almost nobody used it. I normally had a KDA of like 1.15-1.3 in these games, and in MW2 I had a KDA of like 3.3 or something insane. And when I played duo with my brother, despite playing ground war and being only 2/9ths of the team, we had a win rate in the 60s. I was coaching high school lacrosse at the time, and this game was super popular, and my amazing stats thanks to how overpowered and underrated One Man Army was won me all sorts of credibility with my players.
MAG (https://en.wikipedia.org/wiki/MAG_(video_game): This was a PS3 exclusive, and I'm not sure how famous it was. It's simultaneously one of my favorite multiplayer games ever, and one of the biggest disappointments. I love the idea of games with large BATTLES and not just small skirmishes. This game let you play 64 or even 128 players per team, although to be fair the nature of the maps somewhat broke it down into series of 32v32s with some overlap. In fact, the name was originally "Massive Action Game," which is where MAG comes from. That being said, while the large battles were great, the fundamental gameplay was only OK, and there were a lot of design flaws. This came out on the same day as Mass Effect 2, a day which for me will always be known as the day of ultimate hype letdown.
Portal: Awesome puzzle game, but it gets shitty halfway through when you go "behind the scenes." The puzzle becomes more about finding surfaces you can actually stick a portal to than really solving puzzles.
The Total War Series: I will probably just lump all these together and say they are pretty cool.
World of Warcraft was bullshit life cancer with horrible gameplay that I only ever played because my friends did, but I will give them some respect and agree with the others saying how back when the game first game out a long time ago, starting a new character and playing for a little while did feel almost awe inspiring in terms of the scope of the world.