I think you're better off going with specific game examples. Sorry these are old, but any good games with 2 player simultaneous will do (try "let's play co-op" search string ):
Links are to play on youtube. Some of these games have been re-released for the DS, some have not. You said "Nintendo!" Anyway, these are the prototypes of some of the best games I played as a child.
Top down - Overhead
Secret of Mana/SNES
- My #1 recommendation. One of my favorite games. Great cult following to this day. SOM is a fun, longer adventure, so it takes longer to play through, and it's easier to play than Gauntlet (you die/get stuck and have to replay the same levels much more in Gauntlet).
(NES version had 2p simultaneous, DS yet unreleased!) Overhead, 2p simultaneous. I always thought this game was a great archetype of collaboration between 2 players because you really had to collaborate to get through the levels. You have to share treasure and health so the other player doesn't die - having the other player around makes the game much easier for you. This is an arcade style game, the challenge level of this game gets significantly higher as you go through the levels.
[See play on youtube]
Action side scroller:
Typically a bit more violent with action/streetfighting themes.
2p simultaneous here is great arcade/action fun. It's basically beating up the baddies, but the play varies a lot with unique twists on 2p side scroller play for it's time (e.g. level 2, going down the crater, and level 3, with the race cars.)
NES: Double dragon II
Straight 2p side scroller, have fun beating up the baddies. Beating up baddies with your side kick is fun!
Link is to TMNT IV, but II also had a multiplayer simultaneous mode. Again, beating up the baddies. Same mechanic as Double Dragon.
Contra I, II, III all were well known multiplayer simulataneous (this is the SNES contra 3)
Artillery-based 2p action side scroller. You have to share the powerups that drop throughout the level. Having long range firing weapons is a completely different feel than street fighting (ala Double Dragon), you cover your friend by protecting him from enemies with your bullets!
SNES: Darius twin
A more off-beat/less popular one, 2 player simultaneous jet fighter. Similar mechanic to Contra, only you can move vertically as well.
Ok #1: HEALTHY MIND HEALTHY BODY. I'm saying PLAY SPORTS.
KEEP IN MIND A 2P SIMULTANEOUS GAME DOES NOT MEAN FUN TIMES FOR EVERYBODY!! The kids might still fight if one "sucks" or holds the other one back, which is pretty much the cause for arguments on the sports field as well.
What I do is discontinue play for any activity (soccer, water guns, you name it), once they start fighting in a bad (kicking, screaming) way. Tell them to stop, and if they don't stop immediately, then stop the game (until the next day, the punishment has to stick). Once you do that a couple of times, getting them to stop is easier. Once they learn they have to play nice, or don't play at all, usually collaboration follows.