Ridiculous fire sales with four or more games for the same price as the other consoles. Microsoft lost a lot of money just to make sure they could sell that many. And they could afford it.

Microsoft put huge amounts of money into the Xbox. That's another very important point, besides Live, DVD player and Halo. Microsoft basically bought its way into the market with billions of dollars of losses.

In direct competition, the bigger company wins most of the time because it has more resources. That's why making the NX like the other consoles would kill Nintendo. When the Xbox didn't sell well after launch, Microsoft quickly dropped the price and put more money into it. When the Gamecube didn't sell, Nintendo decided to sit things out to avoid losses.

Because of Halo 1 & 2.

Good thing I was in college at the time.  Gamecube provided nothing but local multiplayer fun with Kart and Melee.  My gf at the time was addicted to Kart.

I played a game that was better than Halo and came out a few years earlier.  It was called Starsiege: Tribes.

To be more specific than just saying "Halo" like everyone else... Halo 2 and the success of XBL are what pushed the OG Xbox past the GC. For the first couple of years on the market, the GC and Xbox were practically neck and neck in terms of sales and quality 1st and 3rd party support. Then starting in 2004 support and thus momentum for the GC began to wane dramatically as more and more 3rd parties reduced support or eliminated it altogether for the console. 2005 was even worse for the GC as outside of RE4 there were hardly any major releases for the console all year.

Xbox meanwhile continued to benefit from the advent of XBL and the continued releases and / or ports of major 3rd party titles like GTA and MGS along with console-exclusive PC ports like Doom III and Half Life 2. And then of course when Halo 2 dropped in November of 2004 that pretty much sealed the deal as millions bought an Xbox just for Halo 2's online multiplayer alone.

The Xbox had a couple of my favorite games of all time, knights of the old republic and jade empire... It also had a little game called halo... And it played dvd's... Overall, I think it made a stronger case for purchase.

Personally, I don't get how the ps2 sold so much more as I liked both the GameCube and Xbox more than the ps2. Momentum perhaps?

It hit first, and there was such an expectation on the PS2 it was phenomenal!  Hype alone pretty much killed the Sega Dreamcast. MGS2, Gran Turismo 3 and such were the true defining titles for the PS2 though and yep, as you say, GTA3. That set it going on a path Nintendo and MS could do nothing about.

Xbox Live and with it, Halo (2).

Obviously "Halo". The GCN had the best exclusives of that gen, hands down.

EDIT: It is odd that 3rd party support for the GCN started to wane so significantly in its later years, with the the hardware numbers so neck and neck up until 2004. 

Remember how much Microsoft was losing that console? Crazy that the GCN was profitable right out of the gate.

No. It was definitely better. I am just saying though. It was billed as the place where the multiplats can enjoy the best quality multiplats.