I don't recall this being asked (not in years at least) so here we go.
FPS/TPS - First/Third person shooter games - Halo/Gears of War style
Action/Adventure - Zelda style, Metal Gear
Turn Based Strategy - Heroes of Might & Magic, FF Tactics, Advanced Wars
Real Time Strategy - Starcraft, Warhammer 40,000
Racing - Gran Turismo, Forza
Fight Games - Street Fighter II, Mortal Kombat, Dead or Alive
Role Playing Games - Final Fantasy, Xenogears,
Sports games - WWE12, Madden, MLB: The Show
Simulation Games - Simcity, Train Simulator
Sandbox Games - GTA, Fable, AC (go anywhere you want games)
Platformers - Rayman, Castlevania, Mario, Mega Man (2D games)
Adventure games - The Longest Jorney, Myst
Puzzle Games - Tetris, Portal, Antichamber
MMO Games (World of Warcraft, Everquest, Star Trek Online)
Some games might "kinda/sorta" fit two genres, yes. If you aren't sure what category to place something, follow the examples, but I imagine you'll be fine.
I enjoy a wide range of games, but I'd ask to limit votes to three or four so the whole "vote for everything" makes gauging interest thing ineffective. I think I play everything on that list except racing games. Still love platform games.