Shit... okay, here goes.
First off, checking out some videos is the best way to learn... I'd suggest some of the videos on this site, and also on skiingmotion.com.
A RODEO: Technically speaking, is a backwards-flipping spin; it's often performed like a lincoln loop into a spin. You can usually tell a rodeo by the way a guy sets himself going off a jump- if he sets his shoulders to the side or backwards at an angle, then chances are he's doing a rodeo. Jeremy Weir (rodeonexis) has uploaded some good rodeo videos in the video section here.
A BIO: is, in my mind, technically the opposite of a corked spin. People call it a forward cork... which means you're going off-axis leaning downhill, not uphill. You can tell a bio after the first 180: if the rider is leaning downhill towards the landing, it's bio; if he's leaning uphill, back towards the jump, its corked. Skiingmotion.com has some nice videos of CR Johnson doing bios.
A D-SPIN: can be be performed in any manner of ways. You will most often see it done like a corked 7 that goes almost completely inverted. This means that after the first corked 3 (the rider has completed a full rotation and is facing down the hill, but since he's off-axis, his back is parallel to the ground and he's staring at the sky) the riders performs a kind of off-axis backflip instead of another spin. You can see a d-spin by watching the skier spin first, then go inverted. You should be able to watch a d-spin on this site, on skiingmotion.com, or on any ski video (provided it's not Warren Miller).
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