For the first time in their history, Nintendo had on their hands a third place console (behind Sony's PlayStation 2 and Microsoft's XBOX). But what went wrong here? They did the discs! The discs were supposed to fix everything, right? RIGHT?! The first thing Nintendo did wrong was opt for a tinier disc than its competitors (with the hopes of curbing piracy). It didn't help, and it allowed their games less storage, much like its predecessor's cartridge issue. Not only that, but the smaller size precluded the Gamecube from allowing DVD playback, as the Playstation 2 and XBOX had. Nintendo's always been focused on leaving their consoles as game-focused, but it may have been a little foolish to not realize a lot of gamers liked the simplicity of having their console function as a DVD player as well.
But maybe the real problem was what was on those tiny discs: at launch, the "big titles" were Luigi's Mansion, Wave Race: Blue Storm (AKA "Worse Wave Race 64"), and Rogue Squadron 2. Of these, only Rogue Squadron 2 was worth remembering. The bigger issue was that this was Nintendo's first console to launch without a new Mario game (and once the Gamecube's Mario game arrived, Mario Sunshine, reception was mixed).
At least the controller was pretty great.
The Gamecube ended up selling about 22 million units. Not bad! Until you consider the Playstation 2 sold an astounding 154 million. So...kinda bad.
"The internet? Yeah, like that thing will ever take off. Psh. Next thing you'll tell me is today's youth doesn't want to jam their faces in glowing red eye-destroyers filled with Game Boy graphics!" - someone at Nintendo, probably
Why does Nintendo hate the internet? While Sony and Microsoft were rolling out major online gaming networks, Nintendo stubbornly insisted that no one would be into playing games over the internet, providing half-assed wi-fi solutions to Gamecube and later consoles and handhelds. And the worst part - the Friend Codes, long serial numbers required to find and play with people you know online. Nintendo's goal with their hesitancy and stringent measures was to limit how exposed anyone would ever be to the crude 'n rude world of online gaming by making sure you only gamed with people you knew. XBOX Live, for example, is a notorious cesspool of scum and villainy (also: racism, misogyny, etc.), but it has brought significant dividends along - at this point, more people are using their XBOX 360 for non-gaming purposes, and this is all thanks to Microsoft's embrace of online capabilities. Nintendo had a solid control of their brand, and did not want to give that up. In the meantime, they got left in the dust.
Nintendo seems to be turning a corner with the Wii U, but at this point they're far enough behind that they have an enormous amount of ground to make up for. But with Mario Kart 8's introduction of DLC, all they really need to do is get more 12 year olds yelling racial slurs, and they'll be back in the game.