When light waves are reflected off a conductor, they experience a phase change of π, meaning the reflected wave is inverted, turning crests into troughs. This occurs because the electric field at the conductor's surface must be zero, resulting in the reflected electric field being equal and opposite to the incident field. The phase change does not occur when the reflecting medium can vibrate, such as water waves bouncing off a wall, but only when the medium is fixed. Calculating the amplitude and phase for reflected waves involves understanding these boundary conditions and applying the Fresnel equations for light waves. The discussion emphasizes the importance of the medium's properties in determining phase changes during reflection.