We present the second edition of the Sloan Digital Sky Survey (SDSS) Quasar Catalog. The catalog consists of the 16,713 objects in the SDSS First Data Release that have luminosities larger than M_i_=-22 (in a cosmology with H_0_=70km/s/Mpc, {Omega}_M_=0.3, and {Omega}_{Lambda}_=0.7), have at least one emission line with FWHM larger than 1000km/s, and have highly reliable redshifts. The area covered by the catalog is 1360{deg}^2^. The quasar redshifts range from 0.08 to 5.41, with a median value of 1.43. For each object, the catalog presents positions accurate to better than 0.2" rms per coordinate, five-band (ugriz) CCD-based photometry with typical accuracy of 0.03mag, and information on the morphology and selection method. The catalog also contains some radio, near-infrared, and X-ray emission properties of the quasars, when available, from other large-area surveys. Calibrated digital spectra of all objects in the catalog, covering the wavelength region 3800-9200{AA} at a spectral resolution of 1800-2100, are available. This publication supersedes the first SDSS Quasar Catalog (2002, Cat. <J/AJ/123/567>), which was based on material from the SDSS Early Data Release. A summary of corrections to current quasar databases is also provided. The majority of the objects were found in SDSS commissioning data using a multicolor selection technique. Since the quasar selection algorithm was undergoing testing during the entire observational period covered by this catalog, care must be taken when assembling samples from the catalog for use in statistical studies. A total of 15786 objects (94%) in the catalog were discovered by the SDSS; 12173 of the SDSS discoveries are reported here for the first time. Included in the new discoveries are five quasars brighter than i=16.0 and 17 quasars with redshifts larger than 4.5.