What a tough question. I can't whittle it down really.
Local multiplayer;
Mario Kart 64 - I used to play this all the time with my brother when we were both under the age of 10.
PES 6 - again, I used to sneak in my brother's room and we'd do all nighters playing this game. This game used to have hilarious modifiers like every player is in an ostrich outfit. You also had absolute OP gods like Adriano, Ronaldinho and Henry.
Online multiplayer;
Gears of War - 360 plus Gears of War. First HD console, first truly online console we had. This game is GOAT tier. Not only was the gameplay superb but the community was hilarious. Shit talking, getting on the wall and tea bagging, turkey noises down the mic, Jedi backflip and skydive glitching, gnasher dominating host advantage madness.
Halo 3 - how do you follow up Gears of War? With the greatest FPS of all time (IMO). Traffic cone splatting, throwing stickies a mile in the air and it landing on someone's head, hip firing snipers - this game had it all. Lunacy. Custom games like zombies and duck hunt, racing around on mongooses. Bungie hit it out of the park.
Modern Warfare 2 - just another absolutely perfect game that improved upon the first MW in every way.
Rocket League - really fun game that has stood the test of time (10 years at this point).
Honorary mention;
Mario Kart DS - I think I played this before Gears of War. I showed my brother that we could connect to WiFi and play with other people, it blew our minds.