We report and interpret HST/STIS long-slit observations of the optical and ultraviolet (1150 - 10270 Angstrom) emission-line spectra of the rapidly brightening Spot 1 on the equatorial ring of SN 1987A between 1997 September and 1999 October (days 3869 -- 4606 after outburst). The emission is caused by radiative shocks created where the supernova blast wave strikes dense gas protruding inward from the equatorial ring. We measure and tabulate line identifications, fluxes and, in some cases, line widths and shifts. We compute flux correction factors to account for substantial interstellar line absorption of several emission lines. Nebular analysis shows that optical emission lines come from a region of cool (T_e ~ 10^4 K) and dense (n_e ~ 10^6 cm^-3) gas in the compressed photoionized layer behind the radiative shock. The observed line widths indicate that only shocks with shock velocities V_s < 250 km/s have become radiative, while line ratios indicate that much of the emission must have come from yet slower (V_s < 135 k/ms) shocks. We are able to fit the UV fluxes with an idealized radiative shock model consisting of two shocks (V_s = 135 and 250 km/s). The observed UV flux increase with time can be explained by the increase in shock surface areas as the blast wave overtakes more of the protrusion. The observed flux ratios of optical to highly-ionized UV lines are greater by a factor of ~ 2 -- 3 than predictions from the radiative shock models and we discuss the possible causes. We also present models for the observed H-alpha line widths and profiles, which suggests that a chaotic flow exists in the photoionized regions of these shocks. We discuss what can be learned with future observations of all the spots present on the equatorial ring.
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