No need for brain hurting! I used that terminology because the poster seemed to know what he was talking about, forget the long names. Picture the atom like a tower with multiple barricades. The first barricade is made out of electrons, so when a bunch of electrons are tightly packed, it's hard to get through. The second barricade is made out neutrons which are much more tightly packed and much harder to get through. Electron degeneracy pressure is like saying the amount of force it takes to knock down that electron barricade. When this barricade gets knocked down during a star explosion, the result is called a neutron star, as all that's left is that neutron barricade. Neutron degeneracy pressure is the force it takes to knock down the neutron barricade. When this happens, the rest of the tower goes crazy and builds a hole down in the ground to escape the incoming threats. They build it so deep, that nothing can get out, and this is a black hole.

That was a pretty easy to understand explanation, thanks. Also I'm assuming that force is the state's own gravity, is that correct?

Yes exactly. Those limits I mentioned before, the ones with the funky names before them, refer to how big these things can get before gravity forces them to collapse.