A binary neutron star (or BNS or neutron star binary or NSB, or double neutron star or DNS) is two neutron stars co-orbiting. A BNS could be the remnant of a main-sequence binary star, or could result from a capture. If both are pulsars, it is termed a binary pulsar. A source of detectable gravitational waves is a binary neutron star whose orbit has decayed to the point that the two merge (a neutron star merger).
The terms binary neutron star and neutron star binary are sometimes used to indicate a binary star that includes at least one neutron star. Sometimes the two forms are meant to distinguish between "with two neutron stars" versus "with at least one neutron star" but there isn't complete consistency on such distinction.