Lord of the Rings online. It's incredibly addictive (and free on steam!) I just downloaded it a little while ago though, so I'm still in my noob levels.
I enjoy free to play MMO games. I started playing Kingsofchaos and moved over to Ruinsofchaos. I can't stop checking my account, The game it self is pretty simple and straight forward but it is really addictive. The aim of the game is to build the biggest best army! Stats are opposing and everyone has Attack / Defence and Spy / Sentry. Thats it really, you get an attack turn every 30 minutes and gold depending on how many soldiers you have. You can get more soldiers by clicking numbers or by upgrading your base.
I know I haven't described it brilliantly but anyone even half interested should have a look you won't regret it.
I don't like MMOs, but I like playing all kinds of flash games. Today I've discovered this minimalist point-and-click game called Gods Will Be Watching, and I fell in love with it instantly. It doesn't have much of a replay value (once you win it, of course), but it makes you think and is really quite interesting. You can find it here: http://godswillbewatching.clay.io/
In general, point-and-click adventure games = best games.
Lord of the rings online is the best free online game ever! I also enjoy many facebook games (Not the stupid ones like farmville though). I mainly play paid games now though very few free ones.
Team Fortress 2 on Steam. It's always great to waste a few hours playing it. It's simple and always action packed. I grew up on Team Fortress Classic, so when they made Team Fortress 2 free on Steam I had to jump on it. I recommend it for everybody if you haven't played it before.
I enjoy playing "The Last Stand" series. You play as a guy in a post apocalyptic world infected with zombies. each game has its own goals but basically you travel to cities and towns and search buildings for survivors, food, fuel and weapons. Survivors help you fight the zombies when you are defending your base. Food is used to survive and fuel is needed to advance to the next area. The cool thing with weapons is that you can give them to survivors to help raise your defences. Another game I like is lemonade tycoon. It is fun starting out as a small lemonade stand at the mark and advancing a lemonade empire on busy streets making a decent fortune.