First thing that pops in my head?
Beat-em-up
Streets of Rage series - More than any other beat'em up this series somehow keeps constantly re-emerging in the gaming collective consciousness with the Streets of Rage Remake, legendary soundtrack that keeps getting remixed to this day, and the Beats of Rage mods that keep flooding the net.
Castle Crashers(THE best 2d beat'em up of the past few years)
Dynasty Warriors series - If you live in Japan, beat'em up = DW games
Fighting - The versus series, the Mortal Kombat series, Darkstalkers series
Platformer - Mario, Sonic, random Disney games
Shooter - Treasure games(Take your pick), Contra series, Metal Slug series, PC snobs

No offense guys, but that's what I thought of when I thought of first person and third person shooters
Action - Too broad, my mind draws a blank. This isn't really a genre
Adventure - Also vague, but my thinks of Zelda, Tomb Raider, and random survival horror games
Driving - Mario Kart games, Ridge Racer series, Crazy Taxi, Crash Team Racing
Puzzle - Lucasarts PC games, Crush, Braid, Portal
Rhythm - PaRappa, Space Channel 5, Bust a Groove, Guitaroo Man
Simulation - Um, the Sims

Harvest Moon, Viva Pinata, japanese dating sims,
RPG - Final Fantasy series, Bioware games, Grandia, Skies of Arcadia, Nippon Ichi games
Sports - Smackdown wrestling games, Fire Pro wrestling games, SSX Tricky, NBA Jam, Tony Hawk's Pro Skater, Sega Soccer Slam
Strategy - Starcraft, Kingdom Under Fire Crusaders, Pikmin