What happens when a neutron star collapses?

A normal star is a big ball of gas, its gravity is pulling it together, trying to make it collapse. It's actually held up because it's really, really hot. In the same way that, when a gas is hot it expands, the star's temperature allows it to expand and stay fairly big. 

But when the star gets really old it can explode: what leads to this is that, eventually, it will have burned most of its fuel and it cools down a bit. It starts to collapse under its own gravity. Stars which are massive enough then start to crush the protons and electrons to form neutrons. These form a huge star-sized atomic nucleus, basically just neutrons, known as a neutron star. 

A normal star can collapse into a neutron star. If a neutron star slowly gathered more and more mass then it could collapse again whereby the neutrons couldn't support themselves. It would start to get crushed together and it would get so heavy and dense that it would turn into a black hole.

A black hole is where you get so much mass in one place it distorts space so much that even light can't escape; this will happen whatever the internal structure of the black hole.

We don't know anything about the internals of black holes, and in fact they won't affect anyone outside the black hole, so as far as we can tell a black hole is as far as anything can collapse.