A Neutron Star is formed when a medium mass Red Giant undergoes a supernova. The remaining core does not produce heat through nuclear fusion and so collapses under gravity. Unlike white dwarfs, the mass of a neutron star is too great for the electrons to provide a quantum counter force and so stop the collapse. Instead, the electrons are forced into the protons resulting in an object formed from pure neutrons. These neutrons are sufficiently densely packed to provide a counter force so the collapse is halted.
The resulting neutron star is very small (~2km) and very hot. It also has a huge magnetic field and a very fast rotation since both of these quantities are conserved during the collapse.
The strong magnetic field channels particles outwards from the magnetic poles. When these particle streams intersect Earth the neutron star is known as a pulsar.