This answer expands on @Rex' s answer, so please read it to get the complete picture. It expands on the part elementary particles have in a black hole creation.
It is often said that when the gravitational force exceeds any outward forces or pressures, mainly the electron degeneracy pressure I'm thinking, the star collapses into a black hole. But how can this happen without the Pauli exclusion principle being violated?
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When the simple hydrogen equation does not hold because the potential has been distorted by the gravitational one, an electron can be captured by a proton. This makes a neutron and an electron neutrino. Neutrinos being weakly interacting escape and the neutrons make a neutron star which continues to collapse towards a black hole , if the mass is large enough. There is no problem with the Pauli exclusion or lepton number at this level. Neutrons are composed of quarks which are charged and also obey the Pauli exclusion principle. When the density due to the gravitational collapse becomes large then the whole will turn into a quark gluon plasma. That is as far as elementary particle interactions have taken us. Research is ongoing.
but I don't understand at what point we start our counting of states from the object being a star to a black hole. What happens in the middle? Is there a sharp change?
The point about a black hole is the total mass, such that it does not allow anything to escape from a certain radius. The quantum mechanical behavior from a certain point on is an effective theory joining quantum mechanics and gravitation, a process that is at the frontier of research. It depends on your definition of sharp. Supernovas are sharp.
Supernovae can be triggered in one of two ways: by the sudden reignition of nuclear fusion in a degenerate star; or by the gravitational collapse of the core of a massive star. In the first case, a degenerate white dwarf may accumulate sufficient material from a companion, either through accretion or via a merger, to raise its core temperature, ignite carbon fusion, and trigger runaway nuclear fusion, completely disrupting the star. In the second case, the core of a massive star may undergo sudden gravitational collapse, releasing gravitational potential energy that can create a supernova explosion.