Question

In the context of quantum physics, what does degeneracy actually mean?

Answer 1

It just means you might have multiple orbitals with the same energy. If you've heard of p-orbitals then you know that there are 3 of them, and you can think of there being one on the x, y, and z axis each. Each of these are symmetric in that they can have the same energy as any of the other two. We say that the degree of degeneracy here is 3 because of this.

There's really not a whole lot to this to be honest. It's like looking through your kitchen and finding a bunch of different sized pots and comparing to see that you might have 2 pots that can hold the exact same amount of water so in a sense they're indistinguishable from each other. Except we're talking about orbitals and their energy not pots and water.

Answer 2

different eigen system corresponding to same eigen value ... that is what it means in quantum physics.