Although all pulsars are believed to be spinning neutron stars, not all neutron stars may be pulsars.
When a star becomes a supernova and detonates, it can leave behind a rapidly spinning, dense cinder which is all that is left of its iron core. It spins rapidly for the same reason that an ice skater pulling in their arms spins faster. So, as the ejected gas from the supernova begins to clear out, what is left behind is a spinning neutron star about 20 miles across rotating up to 30 times per SECOND. This body is expected to have a powerful magnetic field, and so has all of the essential ingredients for becoming a powerful accelerator of matter, provided there is matter left over in its surroundings after the supernova. The Crab Nebula in Taurus is the remnant of a supernova that happened around 1054 AD. In the 1960's, radio pulses were detected from a pulsar in the heart of this remnant. Optical flashes were also detected from this pulsar in the late 1970's.
If the star was a member of a binary star system, the companion star could provide the necessary matter, and so we end up with a pulsar in a binary system. The gas is accelerated by the spinning magnetic field of the neutron star, and the charged particles emit 'synchrotron radiation' as they are accelerated to nearly the speed of light. The surface of the neutron star itself can be a source of matter as the neutrons there decay into protons and electrons. These charged particles accelerate out along the magnetic poles of the neutron star and emit synchrotron radiation in pulses. Every time one of the poles of the neutron star crosses our line of sight, we see a pulse of radiation like a lighthouse beacon.
Astronomers have long noted that there are more supernova remnants than their are pulsars, suggesting that not all supernova produce neutron stars and pulsars. There are also fewer pulsars that have been detected, than the expected number given the best estimates for how many stars are likely to be in the proper mass range to produce neutron stars as end products rather than black holes. So, there are still some things we do not understand about how pulsars are formed.