If you put a gyroscope on the end of a hinged bar attached to a base that's free to rotate, with the hinged bar sticking a couple degrees off vertical something very interesting to watch happens. The whole assembly will rotate on it's base as the hinged bar with the gyroscope falls over.
You can also start the gyroscope with the bar horizontal and rotate the base back and forth by hand, and the hinged bar will move up and down.
I don't know where to begin with explaining the physics involved, but the gyrotwister works on a similar concept. As you change the angle (e.g. roll your wrist around) the gyroscope rotates radially (I.E. not along its axis) inside it's spherical casing. The spindle of the gyroscope is loosely fitted into a groove along the inside of the casing, and the radial motion in one axis increases the friction enough for there to be some grip between the spindle and it's guide track, while radial motion in the other axis rolls the spindle along in the guide track, accelerating the gyroscope.
Incidentally, don't be playing with one when you start trying to figure out how it works. You'll space out and when you come back to reality after your eureka moment your forearm will not feel well.
These aren't mere toys, though. My brother used one for physical therapy after he broke his arm.
Also sold under the name PowerBall.