We present details of the construction and characterization of the coaddition of the Sloan Digital Sky Survey (SDSS) Stripe 82 ugriz imaging data. This survey consists of 275 deg^2^ of repeated scanning by the SDSS camera over -50{deg}<={alpha}<=60{deg} and -1.25{deg}<={delta}<= +1.25{deg} centered on the Celestial Equator. Each piece of sky has ~20 runs contributing and thus reaches ~2mag fainter than the SDSS single pass data, i.e., to r~23.5 for galaxies. We discuss the image processing of the coaddition, the modeling of the point-spread function (PSF), the calibration, and the production of standard SDSS catalogs. The data have an r-band median seeing of 1.1'' and are calibrated to <=1%. Star color-color, number counts, and PSF size versus modeled size plots show that the modeling of the PSF is good enough for precision five-band photometry. Structure in the PSF model versus magnitude plot indicates minor PSF modeling errors, leading to misclassification of stars as galaxies, as verified using VVDS spectroscopy. There are a variety of uses for this wide-angle deep imaging data, including galactic structure, photometric redshift computation, cluster finding and cross wavelength measurements, weak lensing cluster mass calibrations, and cosmic shear measurements.