Game Boy:
I'm going to lump in GBC as well, and admit that some of my choices were played way after the Game Boy went off the market. The two games that really nail the feeling of the GB and the GBC to me are Donkey Kong 1994 and Wario Land 3. DK94 is a pocket-sized DK at first that evolves into something even more incredible and content-jammed, which is truly magical in comparison with some of the most iconic GB games, which are, in some ways, poor man's iterations of console-sized games. It's an incredible achievement and one of my favorite games of all time. Wario Land 3, on the other hand, is a lush, colorful game that encapsulates the development of the Game Boy as a whole- Mario Land started as a bare-bones representation of the franchise and spawned something completely new and charming that is regarded just as highly as the franchise it came from.
N64:
Super Mario 64 is undoubtedly the defining title of the N64. It is the joy of movement in 3D perfectly encapsulated in a game. I'd also argue that Super Smash Bros. is the joy of all of Nintendo's IPs in a madcap fighting game, and it defines the platform fighter genre.
GCN:
This is a hard one for me and I have to fall back on two predictable answers- Paper Mario TTYD and Pikmin. I talk about these two games in specific because the GCN was all about either being as weird and new as possible (Pikmin) or it was taking something traditional about Nintendo's classic IPs and giving them a twist (Paper Mario). You took Mario and put him on an island, Metroid turned into an FPS, and Zelda went the opposite direction from realism and darkness (before going right back into it). Paper Mario is Mario, but the corner of the Mushroom Kingdom is really sleazy and the plot is bizarre and varied. It's truly joyful storytelling and RPG.
My honorary third choice would be Final Fantasy: Crystal Chronicles, which is another facet of the GCN that Nintendo tried to push- its weird fascination with the GBA connectivity, but also its multiplayer focus.
Nintendo DS:
I'd argue that Super Mario 64 DS is a defining title because it showed the flip-side of the Game Boy argument- now a console game could be on a handheld, and it could be enhanced. Unfortunately, the ports that would follow wouldn't be as impressive and would largely fall into RPG territory. So, what I would argue instead is that, for better or worse, The Legend of Zelda: Phantom Hourglass. It's a game that reinvents the control scheme for the touchscreen and utilizes the features of the system to their fullest extent. Seriously, the fullest. Extent.
Oh, and since I can, I'll nominate Chrono Trigger DS because it's a port of the best SNES JRPG, which is what the DS was all about.
Wii:
I think about a lot of the stuff I played on Wii and there's this really weird bitterness. Like, there were definitely some things I liked in there, but I can't quite put my finger on them. So, I'll go with three odd titles- Final Fantasy Crystal Chronicles: The Crystal Bearers, Mario Kart Wii, and The Legend of Zelda: Skyward Sword.
The Crystal Bearers is a passable Action RPG title with a fully-interconnected world that has great music, a decent plot, and mediocre motion controls. Throughout the Wii's lifespan, I searched (in vain) for a game that expertly blended motion controls and the kind of gameplay I really enjoyed. Super Paper Mario didn't have it. Super Mario Galaxy didn't have it. All the Sonic games DEFINITELY didn't have it. A lot of what showed up on the Wii was an attempt at adding depth to the simplistic fun of Wii Sports, and there weren't many games that did it proper.
The two exceptions to this would be Mario Kart Wii, a relatively surface-level implementation that was absurdly fun, and Skyward Sword, which was extremely integrated and ultimately the perfect execution for what I hoped motion control would be. I book ended my discussion with The Crystal Bearers and Skyward Sword because it encapsulates the struggle of the Wii- in many ways, The Crystal Bearers is Skyward Sword. It is motion(even in its mini games), character, and world focused, but never on a massive scale.
Of course, Xenoblade would be the alternative to this- a game that pushed the hardware to its extremes and didn't need any motion controls. Ah, well. We can't win them all.
Nintendo 3DS:
The 3DS is arguably Nintendo's best handheld because of its shared library, online store, and enhanced wifi, but I'd argue it is Nintendo's best console since the SNES, as well. It expanded on the idea of portable gaming and sharing those experiences, and that's really what is so special about it. I mean, it also has a fantastic game library, too.
Of course, what games perfectly encapsulate the feeling of sharing with other gamers, while also having pretty decent gameplay, themselves? I can think of no other example that does it as well as Monster Manor, the weird map-filling, ghost-killing title from Prope. What Monster Manor executes perfectly is the feeling of progression and the cooperation regarding your Streetpasses- rooms can be made bigger and net larger rewards when they're built with the same colors, so you want to choose your pieces carefully. The gameplay isn't terribly difficult (not in comparison with Mii Force, for example), but it does favor strategy, and has a fairly impressive sense of progression in its own right. While Find Mii was an exciting taste of Streetpass gameplay, there was often the feeling of "wasted" passes because of the strict enemy limitations and limited character growth. Monster Manor was a more impressive implementation of the concept, and it was all the better for it.
However, we aren't simply talking about Streetpass, as the system also gave players the opportunity to share with one another online. Another example of the joy of the 3DS has to be Pokemon X and Y. While these games introduced 3D in more than a few ways, X and Y streamlined player connection and communication more than ever before with the Player Search System, O-Powers, and Wonder Trade- a joyous addition that was high risk and reward.
One last title I'd like to mention would be Kid Icarus: Uprising. Again, this is an example of Nintendo (or rather Sora) getting the most out of the system- AR functionality, touchscreen gaming, online capabilities, and Streetpass weapon sharing. It's a very divisive title for a number of reasons, but the experiences I had with it were absolute joy.
Wii U:
Ah, and here we are. The Wii U is on its last legs (well, leg, with Zelda not being out just yet), and while I can't rightly say that it is my favorite Nintendo console, it has a number of specific charms that make it memorable in its own right.
First things first, I have to give an honorary mention to Nintendo Land. Adding complexity to the mini-game structure of Wii Sports was a daring move that ultimately didn't pay off for Nintendo, but it struck the precise balance that I so desperately had hoped for with the Wii, unfortunately coming a generation too late. When you have the right crowd, these games are a blast to play- but they really need the right crowd. Captain Toad: Treasure Tracker is another example. In a crunch, Nintendo can still deliver a quality experience- one that offers charm and inventive situations without a huge budget. If they had realized that concepts like this and Kirby and the Rainbow Curse were viable from the start, they could have perhaps buffed their library a bit further. Xenoblade Chronicles X is a massive technical achievement and arguably the best game on the Wii U, but it is very niche and stands alone in its own way.
The true victors in the example of the Wii U are the only games that truly stand on their own in its meager library, those being Pikmin 3 and Splatoon. While Splatoon seems like something that will eventually pop up in portable form, it was unique, bold, and fun, and worked in a way that truly felt like a Nintendo product. Pikmin 3, on the other hand, is a gorgeous, HD Nintendo game that actually benefits from the Gamepad's mapping capabilities immensely, enhancing the core gameplay's strategic elements while helping push the challenge mode to its extremes. These two games use the Gamepad as an interactive map, which is unfortunately the best that Nintendo could muster for the Wii U, but they are fantastic games in addition to that.