In a gymnasium, what is a BOSU ball used for?

A BOSU is a balance trainer. The BOSU is shaped like half of a ball on the bottom of a flat platform. The trainer can be used in two ways. The first is a balance board where the ball allows for movement and requires stability. The second way is with the flat part on the ground it provides a stationary device and the athlete steps on the round part to offer a different balancing experience.

The name initially came from an acronym standing for “Both Sides Up" - a reference to the two ways a BOSU ball can be positioned. It is also referred to as the "blue half-ball", because it looks like a stability ball cut in half. The acronym now favored by the creators is "BOth Sides Utilised"

https://en.wikipedia.org/wiki/BOSU