Imagine the mass of the sun condensed down to the size of about ten miles. The gravity would remain the same but everything else would change. This is in large part what a neutron star is and though our sun is too small to become a neutron star we have discovered these odd bodies and the understanding of them could help us understand a great deal about our universe. A Neutron Star is a star that has collapsed in a supernova into a tiny amount of space.
Neutron stars are created in supernovas as the core of the supernova is condensed down into tiny area. This matter is called Netronium or neutron-degenerated matter if you prefer the more scientific term. This matter is made up almost entirely on neutrons, the particle of the atom that has no charge. Without the electrical charge to force the matter apart gravity is allowed to take full control and condense the matter into neutronium. In fact the only thing that keeps the matter from collapsing even more fully upon itself is the Pauli Exclusion principle that states that no two neutrons can occupy the same quantum state at the same time.
One of the most interesting things about a neutron star is that upon collapse in retains its spin, but because it is now much smaller than it was before it completes a rotation much faster. This is caused by the conservation of angular momentum. Yet this momentum will slow but only at a miniscule rate and on occasion it can "spin up" or undergo a "glitch". This is actually caused by what is called a "Starquake."
Starquakes are caused by the slowing of the star allowing it to become more spherical but neutron stars have a hard crust and just as on earth it must occasionally shift to allow the energy to be released. This reduces the radius of the star and because of the conservation of angular momentum it speeds up.
Neutron stars which emit directed pulses of radiation are called Pulsars. This is caused by their strong magnetic fields. Due to the regularity of the pulses the first pulsars detected were thought to be messages from extraterrestrials and because of this it was LGM-1 where LGM stood for Little Green Men. Pulsars have helped to prove the existence of Gravitational Radiation as put forth in General Relativity.
The study of Neutron stars has helped scientist learn a great deal about the universe and though they have turned out to not be messages from an alien species they are in some ways the universe speaking too us. It flashes the message of the variety and uniqueness of matter throughout the universe and by understanding them we can understand everything a little better.