From what I can gather from responses, we're talking about the DreamCast/PS2/Xbox/GCN era, right? If so, this was not a fun generation for me.
I felt going into this generation, it held promise. Nintendo ditched cartridges and went with discs, so it seemed like they were learning from their mistakes... but of course, they had to design it like a purple lunchbox, which seemed to give some people the impression that it was not even as powerful as a PS2 (seriously, I had to explain this to several people). I also didn't realise that the mini-discs and lack of DVD would be a problem either (didn't care about DVDs myself). The capabilities of the system looked like a good step above the N64, not just in graphics but also in the scope of games.
Then, the system ends up launching without a Mario game, which was the first sign of trouble. I actually like Luigi's Mansion, but it is not good enough to carry a launch. When this Mario game finally comes out... it's got this weird water pack gimmick. It's got less worlds than Super Mario 64. Mario has lost some of his moves, even if he gained a few others with the water pack. This is not what I imagined at all for a SM64 sequel, and it didn't feel like it took any steps forward from that game. I ended up not buying it until years later as a Player's Choice game, and I actually did end up having some fun with the water pack, but I also discovered how unpolished this game was, which was unexpected. Nintendo is better than this.
Zelda is a similar story. We all know the whole deal that felt like a bait-and-switch, but it actually didn't take long for me to get over the goofy style. What ended up bothering me was the ocean thing, as it made this game once again not feel like an increase in scope over the N64 games, even if such a large ocean technically was. I also waited until the Player's Choice before buying this game, and I liked it better than Mario Sunshine, but it's still not a great Zelda and the world actually felt smaller than the N64 games because of so few land masses.
Mario and Zelda are two major games for me, so any system with weak entries in those series is low on my list. I enjoyed Twilight Princess a lot more than Wind Waker, albeit I played it on Wii. I wonder how my perception might have changed had the releases of the two been switched around...
To add to this, the loss of Rare left a noticeable hole in the GCN lineup. Perfect Dark is my fave N64 game and I was eagerly awaiting some sort of sequel. The system still wasn't getting some of the cool games released on PS2, such as its RPGs, though it still saw better support than N64. Some of Nintendo's other games also saw delays, or other such changes I did not agree with. Even though games looked better, most did not feel like a step up from the N64 and PS1 in terms of gameplay, and in some cases felt worse. I felt a lot of waiting around for games to come out that I actually wanted. So much so, I spent most of this generation buying all the N64 games, a system I consistently enjoyed more than GameCube then, and to this day.
That said, the GCN saw Super Smash Brothers Melee very close to launch, a game which was my fave for a long time, and still is one of them. Metroid Prime was also amazing. If not for these two games, I'd likely have sold off the system. However, I've come to appreciate the system more in retrospect, as there are games that I missed for one reason or another that are also good stuff, such as not being able to afford all that many games (another reason I went with N64 stuff, prices were dropping fast). Pikmin is an example, although I played it on Wii so it's sort of a Wii game for me. This is not the kind of game my 15-year-old self would be into, but my adult self thinks it's great.
Though, I think one reason why I appreciate the generation more now is because of how much worse things got after that, and just keep getting worse. I got a PlayStation 2 back at the end of 2011, and collected a nice library so far. It's got some fun games on it, though I still do not regret the fact I didn't own one during its time.
As for the Xbox, I used to own one way back then, but it was stolen. I don't care very much, as I didn't like much on it: my fave game is Fuzion Frenzy, but the Wii serves my party game needs now. Halo was... okay. Serviceable. But most other shooters are better. That's about it.
I rented a DreamCast and knew someone who had one. There wasn't really anything that interested me on the system aside from Crazy Taxi and Sonic Adventure... both later came to GameCube and I now own them there.
Wow. Yeah, did not mean to write that much, but there is the story of Mop it up and Gen VI. And to think this is the short version...