A neutron star is the collapsed core of a massive star. It is what is left of the star, after a supernova explosion.
When a high-mass star comes to the end of its lifetime, its outer layers collapse onto the core. This squashes the star's core to the point where the atoms are smashed apart, leaving only neutrons. Neutrons are sub-atomic particles with no electric charge.
The outer layers are then thrown out into space by a shock wave. This leaves a rapidly spinning neutron star behind. Some neutron stars have been found to rotate at several hundred times a second!
A neutron star can have the same mass as 1 or 2 Suns. However, it will only be about 20 km across. Imagine squashing the Sun until it was the size of a city! For comparison, a house full of neutron star material would weigh the same as the Moon. The only object denser than a neutron star is a black hole.