My abilities as a gamer strongly dictate the genres I like/dislike most, but other things can factor in, too.
The genre I like the absolute best is action/adventure. You're one character in third-person running around and, well, having an adventure. No party-management, usually you fight enemies by pressing a button or two (no having to do quick, precise combos), and there isn't typically a whole lot to juggle/worry about at any given time. Platformers are largely the same, though often more challenging. Throwing in RPG elements like levelling up or item upgrades is cool. Collecting stuff is usually a good feature, too.
Fighting games are a prime example of a genre I dislike mostly because I can't play them, most notably due to that I cannot for the life of me work with that kind of quick-thinking on-your-toes button combo gameplay that is their core. My thumbs don't have that kind of precision and coordination. The same also applies to aiming (or in some cases, even getting around) in shooters.
On the subject of shooters the one other reason I don't really like the genre is simply because I'm not interested in guns or war/sci-fi themes. Throw a fun multiplayer in like in Goldeneye or Borderlands, though, and you've got yourself a shooter I'll try.
JRPGs are the last on my list of top genres I don't like. If it ain't Pokemon, it just seems horribly tedious. I don't want to manage a whole party and all this other stuff and have to worry about the whole group in a fight at the same time. I also really don't like turn-based combat, to be honest. If I'm a guy with a sword I want to just hack away at the monster and get on with it. There also seems to be a lot more strategizing involved than I'd like in your average RPG.
tl;dr
Favorite: Action/Adventure, Platformer
Least Favorite: Fighting, FPS, JRPG
