We provide a quantitative description and statistical interpretation of the optical continuum variability of quasars. The Sloan Digital Sky Survey (SDSS) has obtained repeated imaging in five UV-to-IR photometric bands for 33,881 spectroscopically confirmed quasars. About 10,000 quasars have an average of 60 observations in each band obtained over a decade along Stripe 82 (S82), whereas the remaining ~25,000 have 2-3 observations due to scan overlaps. The observed time lags span the range from a day to almost 10 years, and constrain quasar variability at rest-frame time lags of up to 4 years, and at rest-frame wavelengths from 1000{AA} to 6000{AA}. We publicly release a user-friendly catalog of quasars from the SDSS Data Release 7 that have been observed at least twice in SDSS or once in both SDSS and the Palomar Observatory Sky Survey, and we use it to analyze the ensemble properties of quasar variability. Based on a damped random walk (DRW) model defined by a characteristic timescale and an asymptotic variability amplitude that scale with the luminosity, black hole mass, and rest wavelength for individual quasars calibrated in S82, we can fully explain the ensemble variability statistics of the non-S82 quasars such as the exponential distribution of large magnitude changes. All available data are consistent with the DRW model as a viable description of the optical continuum variability of quasars on timescales of ~5-2000 days in the rest frame. We use these models to predict the incidence of quasar contamination in transient surveys such as those from the Palomar Transient Factory and Large Synoptic Survey Telescope.