I just want to make sure you know that I don't think this is a definitive list of games. This is a list of games that I like, and think should be played. This list is definitely incomplete and may be updated without notice!
Unreal Tournament
Without Unreal Tournament, a lot of other brilliant games would not exist today. Not only is the Unreal Engine responsible for a whole host of 3rd party games and mods, it set a standard of how online gaming should be. Epic MegaGames, indeed they are.
Quake 3
This was the main competition to Unreal Tournament. Vastly improving on the original Quake games, this one was focused on providing a great online experience. It was fast paced like Unreal Tournament, but with more focus on free for all.
Battlefield 1942
The starting point of the Battlefield franchise, this game set a new standard for online warfare. There was both infantry and vehicular based combat. The player capacity was also something to be admired, as it allowed mass team action.
Counter-Strike
Whether you're playing the original, source, or CS:GO, it's clear after a short time that this game will steal many hours from your life. Originally a mod for Half-Life, it was upgraded to a full commercial game and now sets a standard for competitive play.
Ragnarok Online
I played this game a long time ago when it was new. It was addictive and stands out to me because it mixes 2D Anime with 3D locations and functions as an RPG. Looking back, I still say it's good, but probably appreciated more by younger gamers.
DayZ
DayZ is the ultimate zombie survival game. Starting out as a mod for ArmA 2, it is slowly becoming a separate commercial game. It's great and shows how good or bad people can be to each other online. The only downside is it's demanding on hardware.
EVE Online
This game is like Marmite, you either love it or hate it. Commanding ships to fight pirates might sound like fun, but beware... there are no respawns! If you lose your ship then it's gone forever. Dare you venture into the dark depths of null sec?
Borderlands
If the fantastic art style of Borderlands wasn't enough to convince you this is a great game, the multi player definitely should! Shared campaign modes are a lot of fun. There's hours and hours of interesting game play to experience with your friends.
Escape from Tarkov
At the time of writing, this game is a work in progress. It's a first person shooter with role-playing elements such as level ups, perks, quests, and various lore. Set in a fictional region of Russia, factions fight against each other to survive.
War Thunder
At the time of writing, this game is a work in progress and free to play. This is a great game that puts you inside a bomb-equipped air-plane or armoured tank. Battle with other players online and work together to crush the opposing team!
Call of Duty 4
The first modern warfare game in the series! After the epic single player story, the online multi player mode is spectacular. COD4 was the tipping point in the series, setting a milestone as being solid and focusing on game play.
World of Warcraft
Easily the biggest and most successful online RPG ever made. There are a wealth of quests and events in this game. If that wasn't enough, you can work together with your friends to level up as a team or fight other player teams.
Left 4 Dead
One of the best co-op games ever created! Not only a co-op game, but there are zombies! Source engine! Need I really say more? It can be incredibly difficult at certain points, especially if your team mates are novices. This is a real gamers game.
Elite: Dangerous
Much like the original Elite from the 1980's, it's a game in which you can lose yourself for quite some time. Many say this game has no depth, but it's a sandbox, so you create your own experience. If you like flying spaceships, this is worth your time.
ArmA 3
The 3rd installment of the lesser known ArmA series of mil-sim games. Starting back in the early 2000's with Operation Flashpoint, it has progressed to an advanced level in gaming. The possibilities are endless, thanks to the modding capability!