I'm going to edit this post with what I think are good games on the respective systems later.For now here are a few for each system.
NES:Double Dragon series
Good beatemup series.If you want co-op go with number 2 and 3.
Another good beatemup series is Battletoads.It can get difficult at times because of the level design. Battletoads and Double Dragon is also good.
For action platforming I would go with the Megaman series. There is 6 Megaman games on the NES. There are fans of each but generally 2 and 3 are the best in the series by most fans.
There is Ninja Gaiden series on the NES.The 3 games are pretty difficult. Ninja Gaiden 2:The Dark Sword of Chaos is generally the best in the series.
Contra series is great as well. The second one is called Super C. Only 2 on the NES.
Sports.
The NES had lots of Sports games.Here are a few notible ones.
Pro Wrestling,Ice Hockey,Blades of Steel, Baseball Stars, Double Dribble.
Puzzle.
Tetris and Dr.Mario are the first that comes to mind.
Shmups(Shoot em ups)
LifeForce,This game brought forth the Konami Code.
Licence games:
NES had lots of good games that were licence games.Here is a few.
Ducktales,DuckTails 2, Chip and Dale Rescue Rangers,Darkwing Duck,Ghostbusters 2.
SNES:
Platformer:
Joe and Mac.Fun Platformer and has co-op
Super Mario World.
Has good replay value.96 exits can take a while to get but the game can be beaten pretty early on too.
Donkey Kong Country series.All 3 are pretty good.The second is regarded as the best.
Fighters:
Super Street Fighter 2:The New challengers is the most up to date version on the SNES.There is SF2:World Warrior and SF2:Turbo. WW has the inital 8 fighters.Turbo has those 8 and the 4 bosses.SSF2:TNC has 4 more on top of Turbos 12. It also has Turbos speed settings.
Mortal Kombat 2 is regarded as the best in the original series.
With the hit of Mortal Kombat and SF2 the market was flooded with Fighting games.Some were good others were not so good.
Killer Instinct comes to mind first.Introduced the combo system.You could do certain button combonations and your fighter would go into a rediculiously long combo. Had a nice cast of fighters.There were fighters that were easy to use while some were somewhat difficult.
Fatal Fury Special I think had all the fighters in the game from the series up to that point.
TMNT:Tournament Fighters.Great figthing game.I think it has 25 fighters.
RPGs:
This genre is what made the SNES to own for RPG players.
Secret of Mana:3 player co-op had to have a multiplayer adaptor to get 3 players.Playable with just yourself but you need to adjust the CPU AI though.
Illusion of Gaia.Haven't played this but I heard it was good.
Chrono Trigger:One of the best games ever. There was a somewhat recent DS port of the game.It has lots of replay value. 13 endings!
Super Mario RPG. A great game. Squaresoft with Nintendo made this game.
Secret of Evermore was a great game as well.
EVO:The search for Eden is an interesting game.In it you start of as a very basic animal and you eat certain animals to make yourself evolve into different things. It takes place in prehistoric times and you go through the ages as you beat a boss.
The 7th Saga I heard was good but haven't played it myself.
Okay on to Genesis.
I have very little experence with the system so here are a few.
Sonic series. There is 4 games on the Genesis. First 3 and Sonic and Knuckles.
Streets of Rage series.A great Beatemuup series. Great music too.
Madden Football,NHL Hockey 94
Jurassic Park as I mentioned had different gameplay from the SNES,NES, and Gameboy versions. I think it is best of the versions.