Neutron stars are the collapsed cores of some massive stars. They pack roughly the mass of our Sun into a region the size of a city such as Chicago or another large american city. Neutron stars are some of the densest types of massive objects in the universe, at times reaching densities of over 10e14 g/cc. At these incredibly high densities, you could cram all of humanity into a volume the size of a sugar cube, giving one just a sense, even though it is simply impossible to wrap your mind around, the uniqueness of these entities.
They are ideal astrophysical laboratories for testing theories of dense matter physics and provide connections among nuclear physics, particle physics and astrophysics, which have and will continue to lead to incredible innovation in our world. The strongest inferred neutron star fields are nearly a hundred trillion times stronger than Earth's fields, and even the feeblest neutron star magnetic fields are a hundred million times Earth's, which is a hundred times stronger that any steady field we can generate in a laboratory.
These unique stars can and fairly regularly display phenomena displayed nowhere else on the entire planet. These include hyperon-dominated matter, deconfined quark matter, superfluidity and superconductivity with critical temperatures near 10e10 kelvin, opaqueness to neutrinos, and magnetic fields in excess of 10e13 Gauss, only a few of the simply amazing reasons why these nuetron stars are significant to us.