*de-lurks* So many games... And I have a long list of favorites. A lot of the games I love are acquired tastes that I couldn't recommend to non-gamers or those new to gaming.
But if I was to yell at someone "Stop playing that Candy Crush Saga bollocks! Or that Facebook Farmville crap! They're not 'real' games! Play something decent instead!" (except I wouldn't. Whatever floats people's boats), the 4 games I'd use as decent gaming examples would be (in no order):
1) TF2, 'cause it's fun, free, and balanced. It involves strategically co-operating with random people. Only thing is though you'll be playing alongside/against a mixed bag of other players, from elitists to newbies, arseholes to awesome people - If one is new to gaming, that may lead to frustration. But perhaps it's a good introduction to the gaming community. And there's not a lot more I can say about it that Crashpunk hasn't.
2) Portal series (Portal 2 if I have to pick one, it's more refined) - A non-violent puzzle game series with an excellent storyline attached to it, provided you can get your head around it. And Nep already covered this, but I agree with him.
3) Beyond Good and Evil - A fine example of an immersive story-driven game (but could be substituted with Okami, Oddworld: Abe's Oddysee/N'n'T or maybe a Zelda game). Dated classic, but one I've already shown to a few people (mostly children) who weren't into/didn't know a lot about games, and it seemed to really appeal to them. Probably 'cause of the cartoonish setting/graphics, like walking around a painting.
The characters are really well voiced, the game world is amazing, the storyline is deep enough to be interesting while remaining simple and coherent. BG&E's gameplay doesn't have a lot to it, but that makes it easy to pick up.
4)
OpenTTD - Build a transport empire over 100 in-game years. The original
DOS version of this was the first strategy game I got into as a child. It taught me a lot about economics and was very easy to understand, while making me think about how to do things without being too difficult. It was addictive to watch lots of little vehicles traverse the networks that I slowly built for them, increasing in complexity with time.
20 years after the original came out, the open-source version of this is still updated regularly, and the features they've added enhance what was great about the original, while making it stand up incredibly well even against other, more modern strategy games.
(Mute the sound in the vid below if you don't like badly synthesized midi)
I'm bad at reviewing stuff.