Nitrogen (N)-doped carbon materials were shown in recent studies to have promising catalytic activity for oxygen reduction reaction (ORR) as a metal-free alternative to platinum, but the underlying molecular mechanism or even the active sites for high catalytic efficiency are still missing or controversial both experimentally and theoretically. We report here the results of periodic density functional theory (DFT) calculations about the ORR at the edge of a graphene nanoribbon (GNR). The edge structure and doped-N near the edge are shown to enhance the oxygen adsorption, the first electron transfer, and also the selectivity toward the four-electron, rather than the two-electron, reduction pathway. We find that the outermost graphitic nitrogen site in particular gives the most desirable characteristics for improved ORR activity, and hence the active site. However, the latter graphitic nitrogen becomes pyridinic-like in the next electron and proton transfer reaction via the ring-opening of a cyclic C-N bond. This inter-conversion between the graphitic and pyridinic sites within a catalytic cycle may reconcile the controversy whether the pyridinic, graphitic, or both nitrogens are active sites.