@ Words of Wisdom:

Is it possible to play Secret of Evermore with 2 players? You sure??

In addition to the lists posted: Goof Troop for the SNES is pretty fun with 2 players (coop).

IIRC one player controls the boy and one controls the dog.  However the dog isn't always around so it's not perfect.  Much akin to the Secret of Mana games and how it takes a little bit of play time to get the first character.

I can't believe no one is listing Gunstar Heroes for genesis. Shame on you all.

NES
Bubble Bobble/Snow Brothers
Chip & Dale Rescue Rangers
Blades of Steel/Ice Hockey
Gauntlet

SNES
Final Fight/Knights of the Round
Rock n' Roll Racers
Uniracers

Genesis
Sonic 2
Gunstar Heroes
Streets of Rage


N64
Mario Party (first one only)
Jet Force Gemini (although you only get 2nd player after a couple hours it still rocks)
Gauntlet

All of these games are awesome 2 player games and still rock single player if one of you needs to sleep or whatever (with the exception of mario party for single player)

My all time fav coop games are
Super Smash TV
Legend of the Mystical Ninja
Both on Super Nintendo.

Genesis,, only things I haven't seen mentioned
Contra Hard Corp, play the Japanese release unless you wish to suffer greatly
Double Dragon,
Bonanza Brothers
Gauntlet 4 (classic gauntlet but with an enhanced rpg mode, really nice)
Vapor Trail
Cadash
Forgotten Worlds
and a 2 player vs game you might want to check out on genesis
Herzog Zwei, An early if not the first rts game, plus it is a whole lot of fun, just make sure you check out a faq first

and you might want to get some neo geo games,
some really good neo geo co-ops
Nam '75, tough as nails but a good time none the less
Blazing Star and Prehistoric Isle, easy to play, fun to look at horizontal shooters
Andro Dunos, Last Resort, not as pretty and harder but still good horizontal shooters
View Point, someone had the bright idea that co-op shouldn't be the default setting in this one, turn it on in the soft dip screen
Shock Troopers 1 and 2, over head run and guns, multiple paths and a lot of characters
Top Hunter and Spin master, 2 player action-platformers
Metal Slug 1-3 and x, obviously
Sengoku 3, probably the best beat'em up ever, 6 different characters each with their own special moves, a deep and interesting combo system, and it is the best looking beat em up on the system, stay away from all other neo geo beat em ups, they are all ugly clunky and dull
Zupapa and Nightmare in the dark, bubble bobble like games
Ninja Commando and Ghost Pilots fun overhead shoot'em ups,
Aero Fighters 2,3 and Stikers 1945 x, easier and pretty over head shoot'em ups
Baseball stars 2, super baseball 2020, Street hoop, and Neo-turf masters, fast paced arcade style sport games

Here you go:

Snake rattle and Roll (NES) - kick ass game
Ice Climber (NES)
Zombies ate my Neighbours (SNES) OMG, best ever
Secret of Mana (SNES)
Contra 3 (SNES)

that should suffice....

Sunset riders...
You need no more than that game, the ultimate co-op game

That makes no sense at all.