We have investigated the physical conditions in the narrow-line region (NLR) of M51 by using long-slit spectra obtained with the Space Telescope Imaging Spectrograph aboard the Hubble Space Telescope and 3.6 cm radio continuum observations obtained with the Very Large Array. Emission-line diagnostics were employed for nine NLR clouds, which extend 25 (102 pc) from the nucleus, to examine the electron density, temperature, and ionization state of the NLR gas. The emission-line ratios are consistent with those typically found in Seyfert nuclei and indicate that within the inner near-nuclear region (r 1'') the ionization decreases with increasing radius. Upper limits to the [O III] electron temperature (T_e 11,000 K) for the inner NLR clouds indicate that photoionization is the dominant ionization mechanism close to the nucleus. The emission-line fluxes for most of the NLR clouds can be reproduced reasonably well by simple photoionization models using a central power-law continuum source and supersolar nitrogen abundances. Shock+precursor models, however, provide a better fit to the observed fluxes of an NLR cloud ~25 south of the nucleus that is identified with the extranuclear cloud. The large [O III] electron temperature of this cloud (T_e = 24,000 K) further suggests the presence of shocks. This cloud is straddled by two radio knots and lies near the location where a weak radio jet, ~25 (102 pc) in extent, connects the near-nuclear radio emission with a diffuse lobe structure spanning ~4'' (163 pc). It is plausible that this cloud represents the location where the radio jet impinges on the disk ISM.