We present a first-principles calculation of the optical properties of armchair-edged graphene nanoribbons (AGNRs) with many-electron effects included. The reduced dimensionality of the AGNRs gives rise to an enhanced electron-hole binding energy for both bright and dark exciton states (0.8-1.4 eV for GNRs with width approximately 1.2 nm) and dramatically changes the optical spectra owing to a near complete transfer of oscillator strength to the exciton states from the continuum transitions. The characteristics of the excitons of the three distinct families of AGNRs are compared and discussed. The enhanced excitonic effects found here are expected to be of importance in optoelectronic applications of graphene-based nanostructures.