We present the complete sample of protoplanetary disks from the Gemini- Large Imaging with the Gemini Planet Imager Herbig/T Tauri Survey, which observed bright Herbig Ae/Be stars and T Tauri stars in near-infrared polarized light to search for signatures of disk evolution and ongoing planet formation. The 44 targets were chosen based on their near- and mid-infrared colors, with roughly equal numbers of transitional, pre-transitional, and full disks. Our approach explicitly did not favor well-known, 'famous' disks or those observed by the Atacama Large Millimeter/submillimeter Array, resulting in a less-biased sample suitable to probe the major stages of disk evolution during planet formation. Our optimized data reduction allowed polarized flux as low as 0.002% of the stellar light to be detected, and we report polarized scattered light around 80% of our targets. We detected point-like companions for 47% of the targets, including three brown dwarfs (two confirmed, one new), and a new super-Jupiter-mass candidate around V1295 Aql. We searched for correlations between the polarized flux and system parameters, finding a few clear trends: the presence of a companion drastically reduces the polarized flux levels, far-IR excess correlates with polarized flux for nonbinary systems, and systems hosting disks with ring structures have stellar masses <3M_{sun}_. Our sample also included four hot, dusty "FS CMa" systems, and we detected large-scale (>100au) scattered light around each, signs of extreme youth for these enigmatic systems. Science-ready images are publicly available through multiple distribution channels using a new FITS file standard that has been jointly developed with members of the Very Large Telescope Spectro-polarimetric High-contrast Exoplanet Research team.