All stars are composed of a colossal mass of gas which is engaged in a delicate balancing act. On the one hand gravitational forces are constantly attempting to cause the star to collapse; however, as they do the atoms in their core are crushed together under intense pressure and temperature. This causes them to fuse together, releasing a great deal of energy. This energy inflates the star, balancing out the gravity. This stage of a star's life can last for many billions of years.
As a star nears the end of its life it begins to run out of fuel. It can no longer release energy via fusion, and the gravitational forces finally win out and cause the star to collapse. The result is a supernova, a blast which ranks as one of the most powerful natural explosions known to Federation science. The outer layers of the star are blown away in the blast, but the core is compressed so powerfully that all of the atoms collapse in on themselves; the negative electrons are crushed into the positive centre of the atoms which they normally orbit. In a sense, the entire core of the star becomes a single neutral particle - a neutron star.
After they are formed neutron stars tend to spin very rapidly, emitting intense beams of energy from their poles. These sweep the sky like the beam of a lighthouse, and many pulsars can be detected as stars which flash brightly - pulsars. As the stars lose energy their rotation slows and eventually stops altogether, leaving a normal neutron star.
Neutron stars are somewhat common, but pulsars are rather rarer since they do not last very long. The Federation has considerable experience in observing both neutron stars and pulsars; in 2366 the Enterprise-D spent some time at a pulsar when an alien life form abducted Captain Picard from the ship and imitated him to take his place. The aliens were investigating the concept of authority, and to that end the duplicate Picard ordered the crew to approach to within a dangerous distance of the pulsar. The crew declined and the impostor was unmasked.1 Later in the same year the Enterprise-D assisted Dr. Paul Stubbs in his mission to investigate Kavis Alpha. This system consists of a star in orbit around a pulsar. Every 196 years the pulsar approaches close enough to pull stellar material from the surface of the star and onto itself, causing a huge explosion and expelling neutronium at relativistic speeds.2
In 2368 the Enterprise-D was tracking a fragment of a pulsar which was due to pass near to Moab IV. When a colony was discovered on the planet the Starship was able to use a modified tractor beam to alter the fragment's trajectory so as to reduce it's impact on the planet.3
In 2369 The Enterprise-D investigated another system similar to Kavis Alpha; the USS Yosemite had been investigating the system and had become trapped in a streamer of plasma which was passing between the star and the neutron star. The Yosemite had been crippled in an accident, but the Enterprise was able to rescue some crewmembers.4
Other ships have also encountered pulsars. In 2374 Voyager's captain piloted the ship directly between two pulsars which were in orbit of one another, a highly dangerous manoeuvre which she used to force alien ships attached to Voyager's hull to withdraw.5