A neutron star is literally a massive, ultra-dense sphere of neutrons. Like a white dwarf, which is supported against gravity by electron pressure, a neutron star is supported against gravity by neutron pressure. It's easiest to think of this as the neutrons literally touching and pushing against each other (although the quantum-mechanical reality of it is slightly more complicated.)

Neutron stars are the source of pulsars, whose massive magnetic fields and rapid rotation rates fire off pulsing beams of radiation like an interstellar lighthouse. In fact, there's just such a pulsar at the center of the Crab Nebula, which is most certainly the remnant of the supernova that created the nebula in the first place.