Team Fortress 2, the only game you really need, and the world's best hat simulator
Platform: PC, GNU/Linux, Mac
Genre: Team FPS
Released in October of 2007, TF2 is not the oldest game I play and advocate for, but its definitely on the older side of things. It is, however, still one of the better games that exists and one that a lot of people turn to when they can't decide what to play. It has hats, and fun filled promotional videos. What more could you ask for?
Unlike most competitive games, TF2 has a community that is generally friendly and not made of the sort of tantrum throwing types that make most MOBAs a nightmare to play. Whether this is due to the quick nature of the games, the way the teams are autoscrambled when one side wins too much, or just the age of the game, I'll leave to social scientists to sort out. Bu the result is a community that typically ranges from pleasant to meh, I honestly can't recall any time on public servers that I encountered someone throwing a tantrum, or even really being much of a jerk beyond the very passive aggressive jerkdom of being AFK during a fight.
TF2 is a game that is balanced by being unbalanced. There are nine classes, and each has a blend of strengths and weaknesses that keeps any from being OP. A team needs at least one of each class just to make up for the deficiencies each class has, and there's ways for even less skilled players to contribute. There's always a shortage of medics, its a fairly easy class to play (though you will die often, because medics are always a top target), and while following around a Heavy or Pyro you can learn the map.
I would recommend looking at the wiki for at least an overview of a how a class works and what the expectations are for that class, that way when you're playing Pyro you'll know that you need to spy check.
There are a variety of game modes, the classic King of the Hill and Capture the Flag (ahem, sorry, "intelligence") are there, but the busy people at Valve have been steadily adding content, the most recent released only a couple of weeks ago, including new play modes which keep things fresh, though for my money the Payload game is still the most fun. Though there is something to be said for loading beer into an alien UFO to make it crash before the enemy team can do the same.
The only downside to TF2 is that all nine classes are men, and other than the announcer women basically don't exist except as the pinup girls in the locker rooms. But the erasure of women is, regrettably, not something that really makes TF2 stand out, so I'm willing to count it as a good game despite that.
Since TF2 is an older game, it can run on older or less expensive hardware with a nice silky smooth framerate. And, despite being older, it doesn't show its age much, I'd say in large part due to the cartoonish graphics that don't try for photorealism.
More to the point, TF2 is simply fun. It isn't grimdark, it isn't hyper competitive, its just a good time, and one that doesn't take a huge time commitment. But despite a shallow learning curve, there is a fair amount of complexity to the game, but you don't actually have to know the details of when and why you might want to use the Axtinguisher instead of the Frying Pan.
You can load up TF2, hop into a game, and play for several hours, or just a few minutes, and it'll be fun either way, and it is that casual aspect that keeps players coming back when they look at their giant library in Steam and can't decide what to play. For that reason alone, "screw it, I'll play some Team Fortress" may as well be the unofficial motto of PC gamers everywhere.
There are also hats.