GameCube was made by Nintendo, which is an extremely old and experienced Japanese company that existed as far back as the 1800's, when they made playing cards.  With the development of technology, they made their first popular video game in 1980.  They continued to make more and more advanced systems, including the Nintendo Entertainment System (NES), Super Nintendo, and Nintendo 64 (N64), until GameCube came out in November, 2001.  

The most popular game for this system, Super Smash Brothers Melee, lets you play as many of the many famous Nintendo characters (Mario, Link, Samus, etc.) and fight other Nintendo characters.  This is different from an ordinary fighting game because it gives you a lot of maneuverability and lets you string together combos of your own creation.

Star Wars Rogue Squadron II: Rogue Leader is a game with amazing graphics and blazing speed.  It was also one of the very first games to come out for the system.  You start the game playing as Luke Skywalker and finish it as Wedge Antilles.  The point of the game is to fly around in different types of Rebel spacecraft and complete the objectives for each mission.

Probably the most original game of any system, Pikmin lets you play as a captain of a spaceship that crashed on an uncharted planet.  In order to put your spaceship back together, you must order around a bunch of small plant/animal creatures called Pikmin.  The Pikmin can do many things, including building bridges, fighting enemies, and carrying heavy objects.