We report on the observation of the bright, long gamma-ray burst, GRB 090902B, by the Gamma-ray Burst Monitor (GBM) and Large Area Telescope (LAT) instruments on-board the Fermi observatory. This was one of the brightest GRBs to have been observed by the LAT, which detected several hundred photons during the prompt phase. With a redshift of z = 1.822, this burst is among the most luminous detected by Fermi. Time-resolved spectral analysis reveals a significant power-law component in the LAT data that is distinct from the usual Band model emission that is seen in the sub-MeV energy range. This power-law component appears to extrapolate from the GeV range to the lowest energies and is more intense than the Band component both below $\sim$ 50 keV and above 100 MeV. The Band component undergoes substantial spectral evolution over the entire course of the burst, while the photon index of the power-law component remains constant for most of the prompt phase, then hardens significantly towards the end. After the prompt phase, power-law emission persists in the LAT data as late as 1 ks post-trigger, with its flux declining as $t^{-1.5}$. The LAT detected a photon with the highest energy so far measured from a GRB, $33.4_{-3.5}^{+2.7}$ GeV. This event arrived 82 seconds after the GBM trigger and $\sim$ 50 seconds after the prompt phase emission had ended in the GBM band. We discuss the implications of these results for models of GRB emission and for constraints on models of the Extragalactic Background Light. Show Partial Abstract