Here are some examples.
Let's start off with the most obvious implementation: FPS. A genre that drives the PS2 and XBox (and dominates computers) will thrive on the Gamecube. Gone is the fiddling with the joystick. A quick flip of the controller, and you've completely turned around. Aiming is no longer tense; your hand eye coordination will allow you to better attack your enemies using a 3d mouse than with a regular controller (think about how many people are about FPS on the computer.)
Don't like FPS? Let's ignore that and move to a love of the Nintendo community: Zelda. Want to see Link do more than just two directions with his sword? No problem, since you will be controlling his sword. When you swing your arm, Link swings his. When you jab, so does he.
Want to control how tense your bow string is? Pull out the bow and arrow, go into first person mode, and extend your arm. Press a button to lock the start position, and pull back as far as you want.
Zelda isn't your thing? How about some fishing. A whip of the controller and you're casting off. You can bob the line back and forth, left and right.
And the accessories for the controller; you can be sure that these will be fairly inexpensive, meaning that companies can throw in their own little controller to add more depth to the game. How about hooking up the headphone set to talk to your buddies in online games to the controller instead of having to have an entire other attachment to the Revolution?
Now imagine that you hold the controller vertically. You're playing Star Fox. You move the controller, just like in a real jet fighter, and the plane moves with you.
Plus, you have a controller that is in one hand only. This means you can eat cheetos and play games at the same time.
The possibilities for this seem endless. Nintendo was not kidding when they named this the Revolution; we are on the edge of virtual reality, the thing that every geek has longed for since we saw the Holodeck in Star Trek: TNG. What Nintendo is doing is taking the big, hulking interactive setups of yesterday's arcades and turning them into the remote of tomorrow's homes.
You no longer control the machine; you control the character itself. Your arm swing is its arm swing. Your aim is its aim. As we've seen with DDR, gaming is turning into more of a physically interactive medium. With this controller, gone are the days of sitting around on the couch fumbling with the controller. Now, if not standing, you're leaning forward or sitting up straight, slashing with your might or blowing a hole in someone's stomach.
Just imagine if they put out pairs of goggles that really gave you the whole FPS feel.
I think Nintendo has a good chance of winning this round.