How are supernovas and neutron stars related to each other?
This depends on how massive a star is.
This is actually consistent with how massive a star is initially before it runs out of fuel. Lets start with the Star we know and make it a reference so it's easier to understand. A star as massive as our sun, when it reaches its end, its climax won't be as chaotic as more massive stars. The internal pressure is not enough to cater a supernova explosion so it will calmly shed its outer layers and the remnant will be a White Dwarf. A small extremely dense star about the size of Earth but mass of the Sun. The thing to note here is that since fusion has stopped and the gravity still wants to crush the star into nothing, the only thing that's stopping further collapse is the Electron Degeneracy pressure. Since the mass is still with in the range that the Electron Degeneracy can handle, this stops further collapse of the Star.
In case of much massive stars, stars with masses 3 to 8 times the mass of the Sun. When they go off, they go off with a Boom.These Stars have temperatures and masses to burn more and more materials materials heavier than carbon, oxygen all the way up to Iron which stars like Sun couldn't do . When these stars run out of there fuel they end up in a violent supernova explosion, resulting in a Neutron star, a small object about the size of a city. At this point the gravity is so strong because of incredible mass on the core that even the electron degeneracy pressure could not hold the thing in place against the gravity and hence the star further collapses until Neutron Degeneracy pressure takes over and stabilizes the remnant. Hence, a Neutron Star since its backed by the Neutron Degeneracy Pressure.
In case of even massive stars, more than 8 times the mass of the sun. After the supernova explosion even the Neutron Degeneracy pressure couldn't hold the remnant in place and the Star further collapses. It collapses until it reaches a single point in space known as a singularity. This is a Black hole.