Oh geez. Like some others that have posted in this topic, if I were to list all the games I like, the topic would explode. Particularly since I have absolutely stupid amounts of free time on my hands and no particular restrictions on spending money.
.....though I dont buy retail games all that often recently, due to most of them just seeming either crappy or boring. I *much* prefer buying indie titles like AVWW here, or Minecraft, whatever.
For genres though, I'm into a number of them. I do fighting games alot... Guilty Gear, Blazblue, MvC3, whatever.... and I do alot of shmups. Too many of them. If it's a shmup, particularly a danmaku type, wether import or otherwise, I probably have a copy of it somewhere.
I like a good action game as well, though I'm incredibly picky about it. Ninja Gaiden is a favorite.
Alot of retro games as well; got a good collection of 2600 and NES/SNES/whatever titles, so any game that uses some of those old design concepts present in those games will call to me; I see some of that, at least, in AVWW.... wether intentional or otherwise, it reminds me of many of those with all of the platforming and exploration.
And on the note of exploration, that's a concept I'm very fond of. I love Metroid-vania games. In most games I'm NOT a completionist, due to my extreme impatience, but in THOSE games.... yeah, I'll go find EVERYTHING.
And I do puzzle games. Particularly "logic" puzzlers, but I still do love me some falling-block types, or stage-based types (like Toki-Tori).
Oh, and Roguelikes. I love Roguelikes. Dungeons of Dredmor is a big one for me right now. Cant get enough of it.
And I do alot of gaming on my iPad. Found quite alot of things I enjoy on there.
A brief list of some of what I've been playing aside from AVWW here:
Minecraft (always a favorite!)
Dungeons of Dredmor
Everyday Genius: SquareLogic (my favorite puzzler, found on Steam)
Jetpack Joyride (iPad. Absolutely brilliant game)
Dodonpachi Daifukkatsu (import shmup)
Akai Katana (import shmup)
Mushihime-sama Futari (surprise, another import shmup)
Eschatos (yet another import shmup. This one is particularly grand.)
Forget-Me-Not (Indie game. Think procedurally generated chaotic Pac-Man. If you havent tried it, DO SO. It's that good.)
Those are just games I've been playing RECENTLY and often. There's plenty of others that while I'm not currently playing them right NOW, I will go back to them.
And then there's the fighting games, which I'm sorta ALWAYS playing, so I dont bother listing them.
As for what I like in gaming, well... I need challenge, for a start. If a game isnt difficult, chances are I'm bored. And some of the games I play are *very* difficult indeed (Like Futari or Eschatos). I dont like games that hold your hand all the way through. I like exploration, and I like freedom to do things my way. I love games with alot of replay value, and I REALLY love games with randomly-generated everything, like AVWW or Minecraft or DF, as they have ALOT of replay value and I can always, always find reason to go back to them. I also like arcade-y games alot; things where the point isnt to "win", but to get a high score. Q*Bert, Moon Patrol, all sorts of other old arcade games, I play those often.
As for what I *dont* like, well... I dont like too much story stuff in games, or more specifically, too many bloody useless cutscenes. Can thank Square-Enix for my hatred of those. And I'm just too impatient to sit through them, unless the story is VERY well-written and interesting... and they rarely are. If I want story, I'll go read an actual book (and I often do). I dont like long-boring stretches, or things that feel like they were shoehorned into the game and end up serving little purpose, and seeming out of place. And bad controls... I hate bad controls.
Sorry, more typing there than I thought, haha. I tend to do that. But there you go.