A supernova is the explosion, visible on Earth as a apparently new star that appears to brighten over a few days, and then fade. There have been no visible supernovae in our galaxy for several hundred years, and we are well overdue another one, though they are quite often visible in other galaxies.
After the supernova there will be a remnant, a small nebula of hot gas that expands, eventually taking the form of a ring. In the centre of the remnant there may be neutron star or a black hole. A spinning neutron star may be a pulsar.
Supernovae that occur when a old, giant star explodes at the end of its life can produce a neutron star or black hole. But some supernovae form when a white dwarf accumulates matter from another star onto its surface, and then explodes, in which case the star is completely destroyed.
The term "nova" was previously used for all stellar explosions. However the term "nova" is nowadays limited to smaller explosions, caused by the fusion of hydrogen on white dwarf (but no sufficient build up of hydrogen to cause the white dwarf to be destroyed)