Degenerate means having the same energy.
The term degenerate most commonly refers to the electron energy levels. See degenerate orbitals. Additionally, the term is also used to describe the energy level of a molecule. Talking in terms of molecular configurations, each configuration of the molecule generally has a different energy level associated with it. However, in some cases like that of ethane, all staggered configurations and all ellipsed configurations have the same energy respectively and are thus referred to as degenerate configurations.