Neutronium is the informal term for an extremely dense phase of matter which occurs only in the intense pressure found in the core of neutron stars. For reasons that will be explained below, it is not often used in the scientific literature of Astronomy and Astrophysics.
When a massive star creates an iron core whose mass exceeds the Chandrasekhar limit, it will collapse and create a type II supernova. The core of the collapsing star is initially composed of iron supported by electron degeneracy pressure, since the nuclear fusion of iron doesn't release energy. When the core collapses, the densities and pressures in the core overcome even the electron degeneracy pressure and the iron atoms' electrons are compressed into their nuclei where they combine with protons to form neutrons.
physics of material at these high densities is unknown, it is far from clear if the interior of a neutron star is best described as a sea of neutrons. It is possible that rather than a sea of neutrons, the interior of a neutron star would best be modelled as a sea of free quarks or of heavy hyperonss. It is also possible that neutron star material undergoes a number of phase transitions in which the material has radically different properties depending on the density and temperature of the material. It is also unknown how neutron star material would behave if the pressures on the star were suddenly reduced. Because of these uncertainties, the term neutronium is rarely found in the scientific literature.
All of these uncertainties can be summarized in an equation of state which describes the pressure of neutron star material given a certain temperature and density. Calculating equations of state is an active and uncertain area of physics. Frequently in the literature, scientists will refer to a "stiff" equation of state or a "soft" equation of state. A "stiff" equation of state has a higher pressure than a "soft" equation at a given temperature and density.
There is a limit beyond which a neutron star can no longer support itself via neutron degeneracy pressure and would collapse all the way into a black hole. The exact limit depends on the equation of state which is used but estimates range from 1.4 to 3 solar masses. Current equations of state are considerably "softer" than the guesses for equation of states used in the 1970's which had limits of 7 or 8 solar masses. Some theories predict an intermediate form of matter between neutronium and black holes, dubbed strange matter.