My favorite has to simply be RPGs. I don't know about the certain type, but I guess open world is the one that I prefer. The Fallout series remain as my favorite video game series of all time, and I absolutely love The Elder Scrolls and Dragon Age, as well.
My first introduction to RPGs was with Pokémon Ruby. It was, at the time, the only game that I had to my GBA, so of course I played the crap out of it. I loved it, but I got tired of it after quite a while, so I turned to my computer and bought Morrowind and the Fallout Collection. And, oh man, it might've been some of my finest purchases. I honestly haven't completed neither of the first two Fallout games, due to the severe bugs and difficulty, but Morrowind is most likely my favorite RPG of all time. It's become so holy to me that I actually cried at the title screen and music when I decided to install and play it after a few years of just having it sit on my shelf. Morrowind is not without its many flaws, but it is still so very much dear to me.
[EDIT:] I can honestly play any game that is considered old, or retro, if you will. Old games have a special place in my heart.
What was your first introduction to RPGs?