We present the most sensitive ultraviolet observations of Supernova 1987A to date. Imaging spectroscopy from the Hubble Space Telescope-Cosmic Origins Spectrograph shows many narrow ( Delta *Dv ~ 300 km s--1) emission lines from the circumstellar ring, broad ( Delta *Dv ~ 10-20 X 103 km s--1) emission lines from the reverse shock, and ultraviolet continuum emission. The high signal-to-noise ratio (>40 per resolution element) broad Ly Delta *a emission is excited by soft X-ray and EUV heating of mostly neutral gas in the circumstellar ring and outer supernova debris. The ultraviolet continuum at Delta *l > 1350 A can be explained by H I two-photon (2s 2 S 1/2-1s 2 S 1/2) emission from the same region. We confirm our earlier, tentative detection of N V Delta *l1240 emission from the reverse shock and present the first detections of broad He II Delta *l1640, C IV Delta *l1550, and N IV Delta *l1486 emission lines from the reverse shock. The helium abundance in the high-velocity material is He/H = 0.14 ? 0.06. The N V/H Delta *a line ratio requires partial ion-electron equilibration (Te /Tp 0.14-0.35). We find that the N/C abundance ratio in the gas crossing the reverse shock is significantly higher than that in the circumstellar ring, a result that may be attributed to chemical stratification in the outer envelope of the supernova progenitor. The N/C abundance may have been stratified prior to the ring expulsion, or this result may indicate continued CNO processing in the progenitor subsequent to the expulsion of the circumstellar ring.