RPG
Mostly ARPG's and Turn Based RPG, those are the ones I play the most (specially ARPG's).
But my requisites are: a well designed and built character development system (or whatever you wanna call it); engaging quests that support different playstyles and outcomes; fun exploration; good level design; well realized world building; all, or at least the majority, of the different mechanics should be consistent quality-wise (ex.: combat shouldn't be good and the remaining systems unpolished); and lastly if it's going to have C&C, than have the choices being meaningful and consequences impactful enough so that the player gains, or loses, access to content and reconsiders his playstyle.
After that, games with a prominent Action component are the ones I enjoy the most.
Hack & Slash/Beat-em-Ups
However not just any hack & slash games.
I am talking about those with well designed enemies, deep combat mechanics that truly reward player skill and where hitting the enemies feels right.
Such as Dark Messiah, Nioh, Severance, Devil May Cry, God Hand, etc...
Shooter
The kind doesn't matter. Can be something as fast paced as DOOM or as slow as RE4. A FPS or a TPS or even Top-dow (like Hotline Miami, Enter the Gungeon). And long as they are done right.
Platformers
It's a simple genre, but one that harbors so many great games - Super Mario 2/3/World/64/Sunshine, Prince of Persia: Sands of Time, Donkey Kong Country, Sonic, Super Meat Boy, etc...
Action-Adventure
Always tough this tag as bit ambiguous both I can't find anything else to better describe games like Zelda: Majora's Mask or Tomb Raider.
Stealth
There isn't a whole lot to say. I just love the challenge and tension a good stealth game offers.
Metroidvania
Don't think it can really be called a genre, but like the action-adventure case, it's a good tag to describe these type of games.
IMO, I actually consider games like the old RE games and dungeon crawlers to be Metroidvanias.
You explore a big well designed level, or several interconnect levels; you carefully manage your resources; you solve puzzles or complete task to either gain acess to a new level, or a item or some kind of update (these last 2 can give you acess to a unexplored location); there is some optinonal areas/challenges/upgrades to discover; combat is done mostly to drain your resources, change the pacing of the experience and overall increase tension; and finally these games rarely lend a hand or try to guide you.
While it's not genre but a design choice, I am very fond of GOOD open-world games.
Genres that I don't like, but also don't dislike:
Card - Most of the times they have a very close relationship to Lady Luck. Someone that was never very fond of me.
Strategy - Never got into them. So I don't have any real experience in this genre.
Adventure - Don't have a particular reason. But just solving puzzles and reading text isn't enough to truly hook me in. For example I like Portal 2, the game has some some good puzzles and fun story + plus a puzzle editor with loads of excellent user made puzzles but it just manages to grab me for 2-3 days before I move on.
Figthing - Might seem weird considering I really like action games, but the reason is I have very little experience with genre. Melee is the only fighting game I truly understand and love.
Racing - Same reasoning with fighting games. I have very little experience with them, but I do love me some F-Zero GX.
I dislike:
Sports - I may enjoy playing ball with my mates, but sports games are fucking dull to me;
Simulations - I can see their appeal to people that have an interest in race cars, planes, trucks, forklifts... and even realize that they can be useful "training" software. But for me simulators are just dull, even more appealing ones like ARMA.
Walking Sims - They are not games.
Storyfag shit - Just because they are novels that you can interact, it doesn't make them games.