A Rydberg state is a state of an atom or molecule in which one of the electrons has been excited to a high principal quantum number orbital.
Classically, such a state corresponds to putting one electron into an orbit whose dimensions are very large compared to the size of the leftover ion core. The Rydberg states of an atom or molecule are electronically excited states with energies that follow the Rydberg formula as they converge on an ionic state with an ionization energy.
Although the Rydberg formula was developed to describe atomic energy levels, it has been used to describe many other systems that have electronic structure roughly similar to atomic hydrogen.