As the star collapses it must spin more rapidly in order to conserve angular momentum. The Sun rotates once every month or so. A newly formed neutron star rotates about 10 times each second. These objects produce focused beams of radio emission which sweep around the sky as the object rotates. If the Earth lies along one of these directions the beam will sweep across the earth many times per second and we will see the neutron star as a pulsar .
Stars often form as binary pairs.
When the surviving star becomes a red giant, it may dump material onto the neutron star. The material falling onto the neutron star will become extremely hot and will emit X and -rays as it falls onto the surface of the neutron star. Many of X and -ray sources in the sky are likely to be neutron stars with red giant companions. If the mass of the remnant core of a star is greater than about twice the mass of the Sun even the degeneracy pressure of the neutrons is insufficient to support the star against collapse. As far as we know the core collapses to a point characterized mainly by gravity alone.