A neutron star merger (or NS merger), the joining of two neutron stars, becomes likely when the orbit of a binary neutron star becomes sufficiently close that gravitational waves sap energy from the orbit, causing it to decay. When they merge, a gravitational wave event may result, and the ground gravitational wave detectors, LIGO and Virgo can detect them within a certain radius. The sixth LIGO GW detection, GW170817, is ascribed to a neutron star merger, based upon the mass of the merging objects and the GW "sound" of the aftermath.
The result can be a neutron star, or a black hole, possibly after a short (e.g., on the order of one second) life as a hypermassive neutron star.
Like black hole mergers, it isn't clear how the orbits become small enough for gravitational waves to pull them together, and the orbit decay is presumed to take a long time, e.g., more than 11 gigayears for GW170817.