Normal: Let's say you're playing any first-person shooter. There's some part of your setup that will contol the camera, whether it be a mouse, joystick, or thumbstick. If you push it away from you and in the game your character looks up, you use the normal style. This is probably the more commonly used method as most games ship with it as the default setting.
Inverted: You're playing the same FPS. If you push the mouse away from you and in game your character looks down, you use the inverted style. This is the more realistic method, just like airplanes use. You'd expect this to be more popular because of it, but if you play any modern FPS you most likely have to go to the options menu to use this way.
Amidextrous: I'm an amidextrous, but this will need some explaining since I'm pretty sure most people aren't in the same situation:
I played my Nintendo 64 lots as a kid, and the camera was usually controlled with the thumbstick (I mean when looking around such as in Zelda or Banjo-Kazooie, or in FPS games when aiming). The way most people held the controller (including me) left them with the thumbstick under control of the left hand. As it happens, most games used the inverted orientation as their default settings, so I got used to it.
However, then the Xbox comes along with two thumbsticks. Pretty much every game used the the right one to control the camera. Now unless you hold it upside-down when playing, you'll be using your right hand for the right thumbstick (as was I). Most games just so happened to use the normal orientation as their default settings, and I got used to it.
So that leaves me inverted with my left hand and normal with my right one. But what about you?