Stable equilibriums in mathematics and physics consist of situations where the energy at a function minimum is lower than all surrounding points, while unstable equilibriums are surrounded by lower energy points. The behavior of a ball placed on a hill or in a valley provides an intuitive example.
Stable and unstable equilibriums can be intuitively understood by considering the behavior of a ball placed on the top of a hill versus one placed at the bottom of a valley. At the top of the hill, there is a flat point where the ball remains stable and does not roll down the hill. However, if the ball is placed anywhere surrounding the stable point, gravity causes it to roll down the hill and lowers the ball's potential energy. Therefore, the top of the hill represents an unstable equilibrium. Conversely, the potential energy, due to gravity, of a ball at the bottom of a valley is lower than that of the ball at any surrounding point in the valley, so the bottom of the valley represents a stable equilibrium.
Neutral equilibriums are also encountered in addition to stable and unstable equilibriums. These equilibriums consist of regions where the potential energy of a function is equal for some range, but continues to decrease beyond that range. In the hill and valley example described above, a neutral equilibrium can be described as a small flat patch of ground between the top and bottom of the hill.