There are two possible remnants to a supernova explosion. The first is a neutron star. Thus, the star acts light a rotating searchlight, and if you're in the beam of the searchlight, you see an object called a pulsar. If the supernova remnant is less than about 3 solar masses, a neutron star will be the result. Neutron degeneracy will hold up the star, just as electron degeneracy holds up a white dwarf.
Gravity wants to crush the neutrons out of existence, but neutrons are fairly solid, and push back.
However, if the mass of the supernova remnant is greater than about 3 solar masses, the greater gravity will crush even the neutrons out of of existence. If this were to happen, the star would collapse further. In fact, if neutrons are crushed out of existence, then we know of no other force that can resist the crush of gravity. The star will, in theory, collapse all the way down to a point. It will be a black hole.