The equilibrium of a spring is defined as the state when it is in its natural uncompressed position, with no external force applied. When a mass is added, a new equilibrium point is established, which simplifies the analysis of vertical oscillations. This new equilibrium allows the spring to oscillate at the same frequency as it would in a horizontal position, but around this adjusted point. Understanding these concepts is crucial for solving problems related to spring dynamics. The distinction between the natural state and the mass-influenced state is key to analyzing spring behavior effectively.