Some of my favorite genres are:

Roguelikes (turn-based): free ones include the tiles versions of Vulture's Eye/Claw, Dungeon Crawl Stone Soup. Commercial ones would be things like Dungeon Hack or Sword of the Stars: The pit.

Roguelike-likes: this includes games with mostly procedural elements but my favorites are Rogue Legacy, Binding of Isaac, FTL (waiting for that expansion before I pick it up again :D ), and Spelunky. Tower of Guns also falls into this category and I can't wait until it's done :D

Metroidvanias: Symphony of the Night being the most obvious :D I seriously wish there was a PC remake that didn't touch the gameplay at all and just bumped up the graphics and kept the music the same as it was on the PSP/PS One.

Point-and-click adventures: this is actually a new one for me as I discovered I like adventure games after having a Wadjet Eye binge, still need to finish Resonance, though :D

Old school FPS: I like the older ones as they didn't give a damn about being realistic and fun had a priority. DOOM, Duke Nukem, Blood. They have a charm about them that is hard to place. Perhaps it's just nostalgia talking?

RPGs: turn-based ones mostly.