This is a breakdance move that builds off of the basic turtle position, adding one-handed balance and the ever-popular breakdance philosophy of constantly spinning, regardless of what the rest of your body is doing.
Just like in the turtle, the b-boy (or b-girl) faces the ground with legs outstretched in any number of positions, just as long as they don't touch the ground. Balance is maintained as in the turtle, the difference being that only one arm is used for support. The elbow has to come farther in toward the gut area, with the hand basically below the elbow, palm to the ground.
Either arm can be used for support; I find it easier to rotate counter-clockwise on my left arm or rotating clockwise supporting myself with my right arm. I imagine any combination would be possible.
Once the potential handspinner can maintain balance on one hand in this manner for a second or two, it is time to start spinning! Using the free hand, rotation is started. Some people also use their toes to get momentum going. This can work, but IMHO does not look as impressive as keeping the feet 100% off the ground.
When learning this move, it is probably a good idea to use something between your support arm and the ground. Of course, it is always neccessary to have a smooth surface (no carpet), but a hat, glove, knee pad, or even a piece of paper or magazine will reduce the friction between you and the ground enough to make spinning a lot easier.
In fact, when I first started trying this, I thought it was impossible to handspin without some kind of insulator as described just above. Of course, this makes the move rather awkward and very hard to work into real routines. After I developed my handspin for a while, I found that I could spin without anything between my palm and the floor. The key is to put your weight on the base of your hand near your thumb to minimize contact area. I've found it also helps if there is a small amount of dust on the ground and if your hand is a little sweaty, which is fairly easy to guarantee after a little bit of breakdancing.
Anyway, once you can spin around for a while, take your free hand and put it away. Typical positions are on your head or on your back. When you have enough momentum and are able to spin around once, twice, even more... it looks incredible.
Variations of the "basic" handspin include the jackhammer, which is basically a jumping handspin. It is possible, although not common, to think of a 1990 as a handspin with the body oriented vertically over the support hand.