A Degenerate star is type of star which is hold up by the pressure of degenerate matter. It is the generic name of three types of stars, degenerate dwarf (commonly called white dwarf), neutron star and quark star. The first two have been observed to exist in nature, the latter not.
In general in a star the gravitational pull of matter tends to squeeze to star to smaller size. Gravity is counterbalanced by the outward force caused by a pressure gradient, which tends to blow up the star. In general a star is in static equilibrium and both forces cancel each other. In stars like the Sun, or in general in main sequence stars, the pressure is dominated by the thermal motion of atoms of a hot gas.
If the matter consists of degenerate electrons, the degenerate star is called a degenerate dwarf or more commonly a white dwarf. Such stars are observed for the first time in the 19th century. The extreme high densities and pressures are explained in 1932.
If the matter consists of degenerate neutrons, the degenerate star is called a neutron star. The existence of neutron stars is predicted on theoretical grounds in 1933 as a possible state of a large number of neutrons, amounting to about the mass of the Sun. In 1963 neutron stars are for the first time observed as radio pulsar, later also as stellar x-ray source.