I don't know if you have the background to understand the answer, but you can find an "elementary" discussion of it here.http://en.wikipedia.org/wiki/Nuclear_spin
In terms of NMR spectroscopy, you can think of protons and neutrons having opposite spins. If you have equal numbers of protons and neutrons in the nucleus, the spins will cancel and the nucleus will have no magnetic moment. If there is no magnetic moment, the magnet of the NMR will have no effect on the population of spin states, and the nucleus will be invisible to NMR. This includes C12, N14, deuterium, and many oyher isotopes. On the other hand, C13, protons, and N15 for example have different numbers of protons and neutrons, and therefore the population of spin states can be affected by magnetic fields and the nucleus will show a signal in the NMR.