By using the first-principles method based on the density-functional theory, electrical and magnetic properties of graphene nanoribbons (GNRs) with the BN-chain doping are systematically studied. For the zigzag-edge graphene nanoribbon (ZGNR), its multispin-state properties: spin-unpolarized non-magnetism (NM) state, spin-polarized ferromagnetic (FM), and anti-ferromagnetic (AFM) states, are considered. The emphasis on our investigations is the effect of doping position for a single BN-chain. It is found that the BN-chain doping armchair-edge graphene nanoribbon (AGNR) has an increase in bandgap and becomes semiconductors with various different bandgaps upon the doping positions. When the ZGNR at the NM state is doped by the BN-chain, its metallic property is weakened, and the quasi-metallic property can also occur. The BN-chain doping ZGNR at the AFM state makes it change from a semiconductor to a metal or half-metal, depending on doping positions. And the BN-chain doping ZGNR at the FM state always keeps its metallic property unchanged regardless of the doping positions. These results indicate that the BN-chain doping can effectively modulate the electronic structure to form abundant electrical and magnetic properties for GNRs. It is of important significance for developing various kinds of nanodevices based on GNRs.