The large exercise ball is the one you might be most familiar with. They are commonly seen in gyms, studios and chiropractic offices where they go by a variety of names such as stability ball, Pilates ball and Swiss ball. The reason we like to workout on them is that they are a good way to add balance and leverage challenges to a workout. The inherent instability of the ball is useful for core training as the core stabilizing muscles have to work harder to keep you balanced on it.
Mini balls are similar to large exercise balls in that they add leverage and instability challenges to exercises. However, at 9 to 12 inches around, the mini ball is not usually a whole body balance challenge. They are very good for other kinds of stability work. For example, one hand on a mini ball during a push up calls on even more core stability. Or, a seated side bend like Pilates mermaid with one hand rolling out on the ball is quite different than being supported by the floor.
Mini balls also have many uses as props. Holding them between the shins or just above the knees during certain exercises can bring in an extra level of engagement for the inner thighs and feedback into the core.
Some people call the large exercise balls described above medicine balls, but the real medicine balls are the smaller heavy ones that aren't meant to roll. Typically 8 - 15 pounds, medicine balls are used for throwing, catching, and other moves where the momentum of the heavy ball adds to the challenge. Medicine balls have been around a very long time. Wikipedia, an unreliable but adequate source for this kind of anecdote, says that there are references to people training with sand-filled bladders 3,000 years ago.
Small toning balls usually weigh in at 1 to 3 pounds. Using them is not weight lifting exactly but the soft, round shape is nicer to hold than many hand weights and many people feel these balls add a muscle toning component for the arms in certain exercises where there wouldn't be any otherwise. They also give feedback into the core.
These light-weight balls are mostly used in massage-like ways. It can feel wonderful roll your body parts over the balls, adjusting the amount of body weight you release onto the ball to monitor the pressure. Similarly, for release of muscle knots and tension, different body areas can rest on the ball using the body weight to stimulate release in the muscles. Many people have found these balls helpful in working through back pain. A plain tennis ball will often do the trick but the therapy balls are softer and often feel better to sensitive areas.
Rolling out tired feet feels great. Sit with the ball on the floor so you can control the pressure and roll up and down and all around. You can use a tennis ball, a golf ball or a small rubber ball. However, see the nubs in the photo? Those are there to stimulate blood flow and the 7,200 nerve endings in the foot. Once you get used to it.... well, you'll really love it, or not.