A dielectric is basically a material that is a very poor conductor of electrons, so when you place it into an electric field, it becomes polarized (positive and negative charges build up on opposites sides). If the field on the dielectric is too high (above the "breakdown" field) then the electrons break free from their constituent atoms, leaving behind positively charged ions, and the dielectric becomes a conductor. This usually happens in a catastrophic way because the free electrons tend to generate more free electrons and ions as they move.