Ultraluminous X-ray sources (ULXs) are thought to be powerful X-ray binaries (XRBs) and may contribute significantly to the redshift-dependent X-ray emission from star-forming galaxies. We have assembled a uniform sample of 259 ULXs over the redshift range z=0.002-0.51 to constrain their physical nature and their contribution to the cosmic X-ray background. The sample is constructed by crossmatching galaxies from the Sloan Digital Sky Survey with the Chandra Source Catalog and selecting off-nuclear X-ray sources after applying astrometric corrections. The fraction of contaminants is ~30% and shows no evolution with redshift. The host-galaxy star formation rates (SFRs) are systematically elevated relative to the parent sample when matched in host stellar mass. The specific SFRs suggest a slight preference for high-mass XRBs, and the X-ray luminosity scaling relations with host-galaxy stellar mass and SFR indicate that the highest-redshift sources represent relatively luminous XRB populations that dominate their host-galaxy X-ray emission. The fraction of galaxies hosting at least one ULX of a given luminosity increases with redshift over the full range of our sample, as expected if ULXs are preferentially found in galaxies with high SFRs and low metallicities. At z~0.5, the ULX X-ray flux is consistent with the X-ray emission from star-forming galaxies. Moreover, ULXs may account for up to ~40% of the integrated flux from XRBs in the normal galaxy population out to z~0.5, suggesting they may contribute significantly to the overall ionizing radiation from galaxies.