A stoppie involves lofting the rear wheel under hard braking. On sports bikes, they are generally pretty easy to initiate. Other bikes can require a little more effort, or may prove impossible to stoppie. But don't be discouraged, I have a friend who can stand a ZZR250 on its nose. (You may've already noticed this :) )
There again are two types. The Rolling Stoppie and The Stop Stoppie. Again Performance Bikes covered this topic very well in an article entitled "Know Lots About Stoppies".
The Stop Stoppie
A stop stoppie is generally easier to do than a rolling stoppie.
Anatomy of a stop stoppie.
Hard Braking In the inital stage of the stoppie, note how the rider is sitting as close to the front of the seat as possible. The bike is under near maximum braking, and is still traveling forward at around 20km/h. Note also that the clutch is in!
The Lunge Since the bike doing the demonstration isn't exactly over-braked, the rider now throws his weight forward as the bike comes to a complete stop. The timing of this lunge is crucial, although mainly from a gonad-preserving view point :)
Hair Raising And this is the result of getting the timimg pretty much spot on. Make no mistake though, this is one high stoppie. In fact I thought he was going to go over the bars. Never-the-less it was observed by multiple people, and has become somewhat legendary.
The Rolling Stoppie
Rolling stoppies involve lofting the rear wheel while continuing forward progress. The world record for a rolling stoppie is around 400m (held by a mad englishman, I believe (Gary Rothwell?)), so there is certainly some scope for fine tuning your skills. Below is a rough guide. I intend to pad it out a bit when time permits.
Accelerate to around 100km/h (lower gears) , shut throttle, pull in the clutch, progressively squeeze brakes. Continue to squeeze harder and harder until the wheel locks (then stop and buy a decent tire) or the rear begins to lift. Once it starts to lift, squeeze harder - the back'll come up. Modulate the brake pressure to keep it wherever you're comfortable as you come to a halt.
(Of course, what will actually happen the first 20 times you do this is you'll panic when the rear comes up, then you back right off the brakes, and the back thuds back down. But slowly you'll become accustomed to the sensation, and you'll be able to do it from higher speeds.)
The After-Stoppie. (nee. Aftermath)
Perhaps one more point deserves a mention... So, you've come to a stop, the rear wheel is 1 meter off the ground, what now? Well, of course the rear is going to undergo gravitational acceleration. Ideally, this would occur in the same plane as the bike, however it often tries to pivot around the steering head. I have seen a guy fall off his RGV (at a training course, what's more!) after accidently pulling a major stoppie and then having the bike fall to the right. You should be able to counteract this by ensuring that you are in a balanced position on the bike, and correcting any deviation from straight-ahead early on.
OK, I can stoppie. Now what?
Now what, indeed. Unlike the wheelie, there seems a little less scope for flourishing while on the front wheel. Perhaps you should just be content with being able to stoppie past your mates on the freeway. Yeh, that'd be cool :)