A neutron star (NS) is a object basically of neutrons, the result of gravitational collapse of a star after it has used up the energy that created the pressure to resist such collapse. Its mass is large enough that gravity overcomes electron degeneracy pressure and causes additional collapse but not so large as to overcome neutron degeneracy pressure, which prevents it from collapsing into a black hole. Typical neutron star mass is 1.4 to 3.2 solar masses with a radius on the order of 10 km. Stars with lesser mass that don't undergo this collapse remain as white dwarves. Think of a neutron star as roughly "squeezing the Sun so its cross section is about the size of New York City's land area", whereas a white dwarf is more like "squeezing the Sun into the size of Earth". Neutron stars' existence was first proposed in 1934 and the first candidates that have panned out were identified in the 1960s.
The vast majority of detected neutron stars are pulsars (with regular EMR pulses from rotating beams) and magnetars (with erratic pulses), neutron stars that through interaction, produce bursts of EMR that are detectable. Neutron stars are presumed to be extremely hot, from heat generated by the Kelvin-Helmholtz mechanism as they formed, but are so small that they put out very little thermal emission: some nearby ones have been detected with X-rays.
A (central) region within a neutron star can be described as a "mass of neutrons" (neutron degenerate matter), and the density is on the same order as the density of an atomic nucleus. Around it, pressures at different depths produce different states of matter with different densities, with less exotic states toward the surface, which is thought to include iron. Deeper, the material can have superfluid (no friction) and superconducting fluid (no electrical resistance) characteristics. Some theories suggest the center can be a "mass of quarks" rather than nucleons, and/or a density at the center is much as six times that of an atomic nucleus (quark matter, QM or strange quark matter, SQM). The term quark-nova refers to the (theoretical) event of a neutron star gaining this characteristic and releasing substantial energy due to spin-down.
The term proto-neutron star (PNS) refers to a neutron star at the beginning of its life as it goes through some preliminary phases with some slower shrinking, on the order of its first 10 seconds.
The term isolated neutron star (INS) refers to a neutron star that is not part of a binary. As in stars, their evolution is simpler, but with fewer emission mechanisms, making them somewhat lucky finds. Their emission peaks in X-rays due to their high temperature and have been found by X-ray observatories.