In physics two or more different physical states are said to be degenerate if they are all at the same energy level. Physical states differ if and only if they are linearly independent. An energy level is said to be degenerate if it contains two or more different states. The number of different states at a particular energy level is called the level's degeneracy.
In quantum theory this usually pertains to electronic configurations and the electron's energy levels, where different possible occupation states for particles may be related by symmetry. The usage comes from the fact that degenerate eigenstates correspond to identical eigenvalues of the Hamiltonian. Since eigenvalues correspond to roots of the characteristic equation, degeneracy here has the same meaning as the common mathematical usage of the word.
In electromagnetics, degeneracy refers to modes of propagation which exist at the same frequency and longitudinal propagation constant. As an example, for a rectangular waveguide, the TEmn mode is degenerate to the TMmn mode if m and n are the same for both of them.