I think my favorite type of game would be a game that is quite realistic. I do not really enjoy fantasy games. For instance, I really enjoy playing The Sims! I think it is a great game because you get to develop your character they way you would in real life. Also, this game incorporates a lot architecture building, and I love it!
I love story-based RPGs. Honestly, any genre, as long as they have a good story is probably going to draw me in. Dragon Age is really one of my favorite games because of how extensive the lore and story is. It's a beautifully built world and the immersion you feel when playing the games feels... real.
I also really like logic/puzzle games. Haven't played any story-based puzzle games yet (though that sounds really fun!)