What might be left behind in place of the dead star after a supernova explosion?

It depends on the type of supernova. 

• Black Holes. Black holes are caused when the mass of a star surpasses the Chandrasekhar limit, and are so dense that light cannot escape the gravitational field of the star. These stars emit no radiation.


• Neutron Stars. Neutron stars are the cores of stars that have undergone supernova. While, like a black hole, are incredibly dense, they do not have a sufficient gravitational field to turn light around. These stars also tend to have incredible rotation, spinning several times per second. They are almost entirely neutrons, and are packed together in a similar density to that of an atom nucleus itself.


• Supernova Remnant (SNR). Supernova remnant is an expanding cloud of superheated gasses. These can be tens of parsecs across, and can take years to slow down. Eventually, these remnants coalesce into nebulae, where new stars can form. Supernova remnants contain heavy metals that were not present after the big bang. 


• Nothing. Sometimes, a supernova leaves nothing but supernova remnant, with no black hole or neutron star. These are exceedingly rare.

Causes for different supernovas include the initial mass of the star in question, the metallicity of the star (the proportion of the star that is metal compared to the portion that is hydrogen or helium), or the relationship the star may have to a twin star.