Degenerate is used in quantum mechanics to mean 'of equal energy.' It usually refers to electron energy levels or sublevels.
For example, orbitals in the 2p sublevel are degenerate - in other words the 2px, 2py, and 2pz orbitals are equal in energy, as shown in the diagram.
Likewise, at a higher energy than 2p, the 3px, 3py, and 3pz orbitals are degenerate.
And, at a still higher energy, the 3dxy, 3dxz, 3dyz, 3dx2 - y2, and 3dz2 are degenerate.
The number of different states of equal energy is called the degree of degeneracy or just degeneracy.
The degeneracy of p orbitals is 3; the degeneracy of d orbitals is 5; the degeneracy of f orbitals is 7.
We can also compare electron energies. In the following diagrams of hydrogen atom energy levels, the electrons are degenerate.
One of the electrons is spin-up and the other is spin-down. In a non-uniform magnetic field, different spins respond differently; electrons with different spin orientations would no longer be degenerate - they would have slightly different amounts of energy. This is how electron spin was first detected in 1922 by the Stern-Gerlach experiment.
Notice also in these two diagrams that the 2s and 2p sublevels are degenerate. This is the case for hydrogenic atoms and ions - i.e. those with only one electron.