Damped oscillations occur when a system is disturbed from equilibrium and oscillates around its new equilibrium position, with the oscillations decreasing in amplitude over time due to energy loss or dissipation. The oscillations are caused by a restoring force proportional to the displacement from equilibrium, and a resisting force proportional to the velocity of the oscillations. As a result, the amplitude and energy of the oscillations decrease exponentially with time as energy is dissipated from the system.