I have read that what keeps white dwarfs from gravitational collapse is electron degeneracy pressure. How does this pressure prevent further collapse, and how is it related to the Pauli Exclusion Principle?
I'm going to start by going over the Pauli Exclusion Principle. Basically, it says that two fermions (in this case, electrons) can't be in the same quantum state. To expand: No two electrons in an atom can share the same numbers for their four quantum numbers. What are quantum numbers? Well, I'll admit that Wikipedia describes them a lot better than I can, but the important thing here is that no two electrons can have the same spin and energy level. In a neutron star, the electrons are packed very close together, and there's quite a lot of force due to gravity. But the Exclusion Principle triumphs. Electrons near each other must have different energy levels; this leads to energy differences and pressure, which counteracts the force of gravity. Above a certain mass limit (the Chandrasekhar limit), at a stellar remnant weighing 1.39+ solar masses, gravity is too strong, and the stellar remnant becomes neutron star.
I hope this helps.