Rocket League
Would you rather play a football game or a racing game? Why not both? Rocket League is the perfect mash up. It is fast paced and a typical example of a game easy to play, but hard to master.
There are various game modes available, but they all come down to the same objective - score more goals than your opponents. You can play in teams between 1 and 4 players in each team.
What I like about this one is that it doesn't require a lot of time to finish a single match - each one is only 5 minutes long, but it can be extended if it comes to an overtime (usually by 1 or 2 minutes).
Counter Strike: GO
Okay, this one everyone knows, or at least heard something about it. Counter Strike is an online first person shooter with a strong e-sport background. It will force you to make quick decisions every round and often drive you crazy, but hey, it's totally worth it.
The game is usually a struggle between two teams - Terrorists and Counter Terrorists. Each of them got their own objectives and there are basically countless approaches to complete them. The success doesn't depend on skills of individual players, communication and cooperation is the key.
Chivalry: Medieval Warfare
Always wanted to get in the middle of a besiege or cruelly raid a village with a sword in one hand, and a torch in another? Well, here is your chance. Chivalry: Medieval Warfare is a first-person slasher. That's right - first-person, it gives a great feel, especially with good mechanics, which don't take long to work out.
There are several character classes to choose from with various weapons. If you get tired of slashing enemies with a long sword, you can pick up a longbow and fight from a distance.
MTA: San Andreas / SA-MP
Surprised not to see a more recent GTA: Online in GTA V? Personally I find it too problematic - long loadings, connection errors, cheaters.
Legacy San Andreas on the other hand in terms of multiplayer is a masterpiece. MTA and SA-MP have been so well developed by the fans that they gave this game another life. There is still a lot of players online - around 100k in total at anytime.
Both multiplayer mods allow developing custom maps and scripts what lead to creation of countless gamemodes and features. There is something for everyone - deathmatches, racing, free running (parkour), role playing, zombie apocalypse, derby destruction, trackmania-like tracks, police chases - you name it. There is even a gamemode inspired by Rocket League, altough due to old game's limitations, it's far from original.
Currently, up to 2000 players can play on one server.
I know some of you may be offended by lack of your favourite games, but keep in mind that these are my recommendations, I can only speak for what I know :)
No League of Legends? Amazing game with an awful community.
Yeah, sorry, I've heard a lot about it, but just didn't have a chance yet to check it out myself. I guess an awful community is pretty common in a lot of multiplayer games ;)