A neutron star is a very small, dense star composed largely of neutrons and Neutronium, formed from the gravitational collapse of a star's core left from a supernova. Some neutron stars, called pulsars, emit beams of electromagnetic radiation from their poles.
In 2154, Tholians in the mirror universe detonated a tricobalt device within the gravity well of a dead star to create an interphasic rift. This rift connected to the universe of 2268; the Tholians were eventually able to lure the USS Defiant through that rift and into their own universe. (ENT: "In a Mirror, Darkly"; TOS: "The Tholian Web")
In 2269, the USS Enterprise received signals from Questar M-17, a hypergravity object that was probably a neutron star. They discovered an alien ship of great age in orbit about the object, and aboard that ship, a malevolent alien entity that quickly infiltrated every ship system. They were able to escape the entity by tricking it into believing they intended to ram their ship into the star, but seconds after the alien fled, they managed a slingshot, leaving the alien trapped within the star's gravity. (TAS: "Beyond the Farthest Star")
In 2368, a stellar core fragment from a neutron star passed through the Moab system and threatened to eliminate the population of genetically engineered Humans on Moab IV, forcing the USS Enterprise-D to arrange a complex modification of their tractor beam to 'nudge' the star slightly off-course, simultaneously providing the colony with technology that would enhance their existing shields. (TNG: "The Masterpiece Society")