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It can be really frustrating for someone who is computer illiterate to get started playing interesting or fun games, so here's an idea that has been pretty universally accepted:
Play the game Portal.
Even the Escapist's Zero Punctuation loved this game, and for reasons that are so self-evident they don't acknowledge the fifth amendment as legal. It walks you through the process of getting used to how you move around, what the game mechanics do, and how you can go about solving puzzles. Once that brief and excellently worked out introduction is through, you get into some of the most creative and fun gameplay this side of God of War's power vacuum.
It's a great beginner game because you don't have to kill things, be particularly quick with your mouse (until the very end), and it's a thoughtful kind of gameplay that still manages to keep you excited without raising your blood pressure too high. Well, unless you get really frustrated with not winning immediately, but then you shouldn't play Solitaire, let alone a PC game.
The novelty of putting a portal on a wall and its exit on the ceiling never gets old. I could walk through a wall only to fall 20 feet all day. At least, in the game I could. After the first one in real life I'd probably shatter my ankles, shins, feet, toes, or even face. It'd still be fun. One of the achievments in the game is called "Terminal Velocity". You get it by falling through a continuous portal tunnel for 30,000 feet. It takes a long, LONG time, but is still fun to watch.
What's even more fun about Portal is the atmosphere that accompanies the game. Your guide throughout the "testing" facility is a strangely warped computer personality named GladOS. Her misanthropic, off-color, and vaguely psychotic narrative compliments the creepy and warped soundtrack extremely well. You can get the shivers by examining drawings that refer to GladOS left by other "test subjects" before you, or by listening to her talk to you about how "Android hell is a real place.."
Portal is a brilliant piece of technological achievment, a hell of a lot of fun to play, and best of all it can be finished in under ten hours. For casual gamers who only play about half an hour a day, that's ideal. All in all you end up with a game that is pretty much perfect in every way.
Unless of course you hate puzzles and only want blood and gore. Then you can go play Quake or something.