I love games that have an atmosphere, visually, musically, through writing, through acting or general quality of design, that draws you in, that feels like a whole world exists in there. Doesn't matter if it's a platformer, sci-fi RPG, fantasy action, FPS, I'm hooked on really great and in-depth world design. And I don't mean "the world map", I mean the existence of everything in the game, foreground to background, character to lore, items to interactions.
On the flipside I love the occasional pick-up-and-play arcade games where high scores and self-improvement are the goal, or multiplayer minigame shinanigans.
And I adore a well crafted co-op/teamwork game in almost any suit.