Neutron stars and black holes
Neutron stars are leftover cores of massive stars that have undergone core-collapse supernova events. In the image above, the neutron star resides in the white-hot region in the center, and the material surrounding it is what is left of the rest of the star after it blew up.
As is evident in the above image, a neutron star is typically born with a kick, sending it out of the center of the debris leftover from the outer parts of the star. The supernova event is not symmetric, so the explosion pushes the core aside. The velocity of a neutron star may be 100 km/s.
While the neutron star is within the supernova remnant, the material can be gravitationally attracted to it, forming an accretion disk. The dust and gas of the disk rotates around the neutron star and accretes onto the star. The inner disk material can become very hot and eject away from the neutron star in jets columnated by the strong magnetic field.
This time lapse video of the Crab pulsar shows a the effects of the rapidly rotating neutron star interacting with the surrounding plasma. The waves moving through the plasma are largely caused by the intense magnetic field of the neutron star. This movie was made from 24 observations made by the Hubble Space Telescope in 2000-2001.
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Neutron stars rotate very rapidly, with rotation periods of a few milliseconds to a few seconds. The combination of the rapid rotation and the beaming of light from the magnetic axis, which is offset from the rotation axis, produces an effect called the "pulsar" effect.
This animation illustrates how a neutron star generates a pulsar signal. This neutron star is gravitationally capturing matter from its red giant binary partner. The matter forms an accretion disk, with the matter near the inside edge of the disk streaming onto the neutron star. The hot matter streams out in jets along the magnetic field lines near the magnetic field axis. Since the magnetic pole is not aligned with the rotation axis, the jets rotate around like light beams from a lighthouse.
Go to the Astronomy Interactives website link above to explore the relationship between the magnetic axis offset, the spin axis offset and the lighthouse effect. The magnetic axis offset is the degree of offset between the magnetic field axis and the rotation axis. The spin axis inclination gives the alignment between the neutron star rotation axis and the Earth's rotation axis. We use the term "pulsar" to denote neutrons stars that appear as repeating flashes from Earth. As you can see, even if all neutron stars beamed jets of light from their magnetic field axes, some of them would not be oriented correctly for us to see them as pulsing light signals.
Image credit: NASA/CXC/M.Weiss Image source
A neutron star has very tightly packed neutrons in its interior and a crust made of solid iron. A neutron star is very much smaller than our sun, or even the planet Earth. The diameter of a neutron star is only about 12 miles across. The mass of a neutron star is about 1.5 times the mass of the sun. This means that a neutron star is extremely dense.
Some physicists think that a closely related object exists, called a strange quark star. Recall that in the late stages of a massive star, the main pressure in the core, working against the pull of gravity, is neutron degeneracy pressure. If the mass of the progenitor star is high enough for gravity to dominate the neutron degeneracy pressure, but not high enough for the core to collapse into a black hole, there may be an intermediate object formed that is supported by quark degeneracy pressure.
This NASA video describes and illustrates the discovery of a binary system including a pulsar. What they saw was a star that brightened and dimmed as it orbited around the common center of mass between it and a pulsar partner.
When pulsars were first discovered, they were called LGM objects, for Little Green Men objects, because it was thought that the pulsing signals could be communication from beings from outer space. This video from History.com offers a good introduction to the discovery of pulsars and quasars, and explains how the pulsed light is created.