Pulsars where discovered by Jocelyn Bell and Antony Hewish in 1967. They were using a radio array to study the scintillation of quasars. They found a very regular signal, consisting of pulses of radiation of a few seconds. Terrestial origin of the signal was ruled out because the time it took the object to reappear was a sidereal day instead of a solar day. The original name for the object was "LGM", Little Green Men, thinking of it as a beacon made by some extraterrestial intelligence. After more speculation, an agreement was reached that the only natural object that could be responsible was a neutron star, a kind of object up to then only hypothesized.
In the 80's a new discovery was made, the millisecond pulsars, that, as their name indicates, instead of having periods of a few seconds, have periods of a few milliseconds.
Also important was the discovery of a pulsar in a binary system. The high precision of the measurements allowed astronomers to calculate the loss of orbital energy of the system, which is thought to be emitted as gravitational waves.
There is general agreement that what we observe as a pulse is what happens when a beam of radiation points in our direction, once every rotation of the neutron star. The origin of the beam is related to the misalignment of the rotation axis and the axis of the magnetic field of the star; the beam is emitted from the poles of the neutron star's magentic field, which may be offset from the rotational poles by a wide angle. The source of energy of the magnetic field is the rotational energy of the neutron star, which is slowing down.
Millisecond pulsars are thought to have been spun up to high rotational speed by infalling matter pulled off of a companion star.
Of interest to the study of the state of the matter in a neutron stars are the glitches observed in the rotation velocity of the neutron star. This velocity is slowing down steadily, except by sudden variations. These were for a time believed to be "starquakes" due to the adjustment of the crust of the neutron star. Models where the glitch is due to a decoupling of the possibly superconducting interior of the star have also been advanced.
As mentioned above, the discovery of pulsars allowed astronomers to study an object never observed before, the neutron star. This kind of object is the only place where the behaviour of matter at nuclear density can be observed (though not directly). Also, millisecond pulsars have allowed one test of general relativity in conditions of an intense gravitational field.