I have written similar posts like this on my now dead blog. I love these kinds of challenges.
2D Platformers
If you want to sell someone on 2D platformers, I always pick Super Mario World. It has delightful character design, colorful levels, mostly happy music, the level design is great for beginners and just challenging enough for veterans. The new mechanics it introduces have a long lasting effect on platformers as a whole not just Mario games. I also like how it is the first to use a save feature that is not tied to specific save points but does require you to actually earn your save by completing a specific type of level. The music in this game is the best of the genre too, the only Mario game where I think the music is actually good enough to just sit and listen to for no reason other than to put a smile on your face.
3D Platformers
Normally I would say it is a toss up between Super Mario 64 on N64, and original Crash Bandicoot on Playstation. I like Mario 64 because it basically creates the template for 3D platformers and is a lot of fun. I dislike the game because honestly once you unlock the upper level you realize how short, and limiting in scope the game really is. It is a great pick up and play game to initiate a gamer into 3D controls as well. The reason I would suggest Crash Bandicoot instead would be because it is a good blend of 2D and 3D and it gets both right. The 3D is not quite as open ended as Mario 64 but it still works really well for what it is plus it has the benefit of not rehashing the same levels over and over. The challenges are quite fun and the game is remeniscent of DKC on SNES, but it feels like a proper evolution of that game more so than DK 64 even did.
2D fighting games
For someone new to fighting games but looking to get a real experience I would recommend Samurai Showdown for Neo Geo because it has all the elements of a good 2D fighter but it is simple enough for a beginner and advanced enough to lead into the superior games down the road. Otherwise I would probably suggest Marvel vs. Capcom as a good entry point for someone looking to jump right into a fully fleshed out fighting game. Anything newer than that is going to be too complicated for a beginner and I don't think Mortal Kombat or Street Fighter cut it because they are fine games just they rely on unique gimmicks that are not quite as indicative of fighting games as a whole.
3D fighting games
I suggest Tekken 3 as a good entry point. I would also suggest Dragon Ball Z Budokai PS2/GC if they were anime fans but Tekken 3 is a really good starting point for 3D fighting games, that is the point where the genre finally caught up and surpassed 2D fighters, before that 3D fighters were all training wheels for Tekken 3.
Scrolling Shooters (shmups)
Everyone is going to have their own opinion here, but I would suggest starting with Raiden, and go from there. Raiden is more accessible than the more advanced shooters, it has a good assortment of levels and weapons but it isn't too complicated like the more popular games and its got some pretty cool music.
FPS
Quake on PC. Doom is the grandaddy but I think Quake is the game that took multiplayer FPS games to a new level. You could pick anything newer as well but I think Quake is where you need to start.
Puzzle Games
Tetris on Game Boy, the definitive version of Tetris. This game is addicting and will turn anyone from a casual gamer to at least a hard core Tetris nut for a while.
Sim/godlike
A tough one. My personal preference is Sim City for SNES, I play that game way too much. But it is far from accessible and hardly resonates with non-gamers or new gamers. I would then go instead with Civilization for PC instead.
Slasher/hack n slash
Golden Axe for Arcade (or Sega Genesis but realistically that won't fly in todays world)
This is the game I think of when I describe a hack n slash game.
Maze runner
Gauntlet hands down no need to explain.
gaming in general, Pac-Man, EVERY gamer needs to play this game first it is like gaming 101. There is no other video game that transcends gaming it is so universal its not even a contest Pac-man exists on just about every game machine ever made in some form or another.
Racing
Daytona USA arcade. I played this game so much in arcades. not as much as Pole Position or Super Chase HQ but those games are incredibly dated and primitive at least with Daytona you get a fully fleshed out experience. There are better games but this is the one I think of it is my go to racing game.
RPG
You are going to hate me for this but seriously Final Fantasy 7. That is the game that made RPG gaming cool and mainstream. As a introduction to RPG games I would dig Dragon Warrior on NES that is damn fun but way to hard core for modern gamers or gamers not already familiar with gaming. FF7 is the ONLY RPG game I know of personally people who have played, enjoyed, and beaten it who literally it was their only RPG experience. Every other RPG appeals more to dedicated fans of the genre.
Dungeon Crawler/Adventure quest
Legend of Zelda NES, or Link to the Past. NES is my personal favorite and the one I think really introduces gamers to the game play but Link to the Past is the better game and the one that has connections to the modern games and is more accessible to today's games in that it feels similar to the modern hand held games.
Brawler/beat em up
I can't decide between Battletoads, Streets of Rage, or TMNT Arcade game, they all resonate with me for the genre. Battletoads is more sci-fi and features a variety of game play but so does the Turtles games. Streets of Rages perfected the formula that Final Fight and Double Dragon began awkwardly. If I had to pick just one, the Sega fan has to hold his lip and go with TMNT.
Button masher/arcade
Ghost N Goblins, this game will frustrate the hell out of you but it really is a great beat the **** out of your controller game.
That is all I can think of at the moment. Not very insightful I am afraid.