How is it that some neutron stars become pulsars and others do not?
In order to answer this question, it is necessary to understand the relation between neutron stars and pulsars. A neutron star has a very powerful magnetic field (about 1012 gauss compared to about 0.6 gauss on Earth) and spins very fast (about 100 times a second). Further, there is a plethora of charged particles in the environment of a neutron star, so that the neutron star emits jets of radiation through the magnetic poles. You may be aware that the magnetic pole and geographic pole (axis of spin) do not coincide in Earth. Similarly, they do not coincide in several neutron stars. So, when the neutron star spins, the beams of radiation are swept around the spin axis. If we happen to lie in the path of the beam, then we see a pulsar. In many cases, Earth does not happen to lie in the path of the beam, and so we do not see a pulsar.
Edit by Michael Lam on August 29, 2015: We now know that not all neutron stars, even the ones we can see because a beam passes our line-of-sight, are pulsars. Some are known as magnetars, with even more powerful magnetic fields. The radio beam is powered by the decay of this magnetic field rather than the the slowing of the rotation of the pulsar. It is not known exactly why some supernovae result in pulsars and some result in magnetars.