I prefer games that have a definitive "end", possibly with some sort of meter that tells me how much I've completed the game. I like games telling me what to do and fulfill each and every objective/mission/challenge it throws at me.
Also, I have a hard time sticking with "open" games that go on for as long as you're willing to play it, such as Minecraft. It probably wouldn't be a problem for me if I could access the online aspect of gaming, but unless the infrastructure of my country's internet suddenly gets a lot better or I somehow get rich quick, that aspect of gaming is mostly out of my reach. I'm not much of an online gamer, anyway.
I'm not very competitive when it comes to gaming with other people. I'm usually up for it if someone wants to play against me, but don't expect me to gloat if I win or get upset if I lose.
Oh, and I flat out refuse to play games where people can just buy their way into getting "better" at the game, such as most MMOs or social/mobile games. Because fuck those types of games.