Based on density functional theory and first-principles method, we investigate the structure and the electronic property of graphene nanoribbion with width N=8 and with or without hydrogen saturation on their edge. Our results show that the carbon atoms on the edge of armchair graphene nanoribbon without the hydrogen saturation are bonded together by triple bonding, which is stronger and more sensitive than that in the case of hydrogen saturation. This type of graphene nanoribbon can serve as a kind of basic material for nano-sensor. Our band structure calculations indicate that both armchair and zigzag nanoribbions are of semiconductor possessing an energy gap. Furthermore, the energy gap of nanoribbon without hydrogen saturation is larger than that with hydrogen saturation, which implies that hydrogen saturation has distinct decoration to the property of the nanoribbon. By the calculation of the paramagnetism, ferromagnetism and anti-ferromagnetism states of the zigzag graphene nanoribbon, we find that anti-ferromagnetism state is the most stable among them, and its magnetism on the edge is strongest, which is suitable for the application in spinelectronics.