Dyna-Flex International (http://dynaflex-intl.com) is the manufacturer of the PowerBall listed above. For specs on the different styles of PowerBall available, please refer to that site. For generic information, check out gyroscope, gyroscopic motion or gyroscopic stability. For specific physics of the PowerBall, keep reading!
Angular momentum is the quantity in question when dealing with gyroscopes. I don't remember the exact calculations, but every object has an 'angular inertia', usually labeled I; then the angular momentum p=I*a, where a is the angular velocity. This is the groundwork.
Next up is torque, or 'angular force'. This quantity is equivalent to the angular acceleration times the angular inertia. Torque can be achieved in several ways, but the most interesting (related to the PowerBall) is the following: when an object has angular momentum, and a torque t1 is applied in a direction perpendicular to the object's angular momentum, a second torque t2 is generated (by the cross product of the angular momentum with t1). This second torque is perpendicular to both the angular momentum and the direction of t1.
The PowerBall is a gyroscope with a ring around its middle (perpendicular to its motion). The ring is relatively free to move within the PowerBall's structure, though there is a little friction. So the gyroscope has two directions of angular freedom. Now suppose that the gyroscope has an angular momentum a, so it is spinning (in the gyroscope direction, call the direction g). Suppose that a torque is applied in the non-free direction (the direction perpendicular to the gyroscope and its ring, call the direction nf). Then a torque is generated in the ring direction (call it r). But there is friction in the r direction, so the cross product of the actual torques t1 x t2 is something other than the angular momentum a. The point of the PowerBall is to time the forced torques (in the nf direction) just right so that t1 x t2 is greater than the previous angular momentum a, which will make the gyroscope portion spin faster.
At its top speed, the 'heaviest' PowerBall (the one with the greatest angular inertia) will generate about 38 lbs. of torque (this is at about 9000 rpm, see the website mentioned above for more details). I recommend this 'workout toy' for anyone and everyone; it's a lot of fun!