A hypermassive neutron star (HMNS or supermassive neutron star) is a neutron star with sufficient mass to collapse into a black hole, basically, more massive than the Tolman-Oppenheimer-Volkoff limit. They immediately collapse unless their rotation sufficiently reduces the effect of gravity. (The pair of terms have been used to distinguish these cases: hypermassive if it is fated for immediate collapse, and supermassive if rotation is sufficient.)
They are of interest because they may exist for a short time in the midst of a neutron star merger, and their short existence can affect the observable phenomena, both electromagnetic radiation and gravitational waves, and may in fact be a source of some as yet unexplained phenomena, e.g., some gamma-ray bursts. GW170817's merger result followed this course.
They are also of interest because of the potential that some might persist due to rotation, which may have the potential of explaining other observations, including transients that might occur if they eventually collapse.