What is a neutron star?
Short answer: A small, dense star made mostly of neutrons, formed from the leftovers of an exploded star (supernova).
Common types of star
Long answer: A neutron star is the leftover core of a dead super-giant star. When the largest stars reach the end of their active life and have no more usable fuel, they undergo an enormous explosion called a supernova. Following this, they can either become a neutron star or a black hole.
Neutron stars are small—typically around 20 km (12 miles) in diameter. However, because they are so dense, they have about 1.4 times the mass of the Sun. A teaspoon of material from a neutron star would weigh about 10 million tons.
The density of neutron stars is caused by the amount of matter collapsing under its own weight. While a star is still active, nuclear fusion in its core pushes outwards and stops the star collapsing. Once there's no more fuel and fusion stops, gravity takes over and the star's core collapses. The gravity is so strong that atoms collapse, with electrons and protons combining to form electrically-neutral neutrons.
If there is enough matter in the collapsed star's core, it will continue collapsing past the neutron star phase and become a black hole.