Great questions! The type of ball that Ryno was on is commonly referred to as Fit Ball; however, don't get caught up on the name, but rather the type of ball it is. Essentially it is a big bouncy ball that comes in several sizes depending on your height (there is a scale on the side of the box as to which size ball you should purchase).
Regarding standing on the ball, this is a combination of functional balance (scientifically referred to as Proprioceptive balance), core body strength (abs-all layers, gluts-butt muscles, and lower lumbar) and mental focus. You can "teach" yourself to do this but you will have to do it in stages.
Establish stablility & coordination within your pelvis.
You can begin by standing on one leg with your eyes closed for 30 seconds. You will feel how the lower leg, then pelvis all strive to create balance. Notice how much you "move" and you are standing on a solid surface. As you progress from the ground, move to an Indo-Board - first with two feet and then to one foot with the Indo-Board on the Indo-pillow.
As your stablity improves, add some movement to improve the strength of the muscles (primary and secondary) that allow for movement. If you are standing on one leg, your leg muscles, glutes and lower back have to "tighten" to keep you from falling over. If you now bend over (without weight) while standing on one leg, you will force the muscle tissue to strengthen to complete the range of motion.
Add Speed/Weight to your movement
After you can move your shoulders, arms, legs, etc. without falling over, you can begin to add weight to the movement (weighted medicine ball, dumbbells, etc.) and then you can slightly increase your speed of movment with light weight.
NOTE: YOU GAIN YOUR STRENGTH WHEN YOU INITIATE MOVEMENT AND WHEN YOU DECREASE MOMENTUM. THIS PUTS THE GREATEST LOADS ON THE MUSCLES DURING THIS TIME. HOWEVER, THIS IS WHERE MOST ATHLETES GET INJURED BY DOING TOO MUCH WEIGHT AND/OR MOVING TOO FAST - ERR ON THE SIDE OF CAUTION HERE.
Once you have mastered all three of these steps, you can see how standing or even kneeling on a fit/swiss ball is a realistic goal; however, it takes a lot of functional strength to make this happen.