We report the results of a ${sim}4$-year direct imaging survey of 104 stars to resolve and characterize circumstellar debris disks in scattered light as part of the Gemini Planet Imager Exoplanet Survey. We targeted nearby (${lesssim}150$ pc), young (${lesssim}500$ Myr) stars with high infrared excesses ($L_{mathrm{IR}} / L_star > 10^{-5}$), including 38 with previously resolved disks. Observations were made using the Gemini Planet Imager high-contrast integral field spectrograph in $H$-band (1.6 $mu$m) coronagraphic polarimetry mode to measure both polarized and total intensities. We resolved 26 debris disks and three protoplanetary/transitional disks. Seven debris disks were resolved in scattered light for the first time, including newly presented HD 117214 and HD 156623, and we quantified basic morphologies of five of them using radiative transfer models. All of our detected debris disks but HD 156623 have dust-poor inner holes, and their scattered-light radii are generally larger than corresponding radii measured from resolved thermal emission and those inferred from spectral energy distributions. To assess sensitivity, we report contrasts and consider causes of non-detections. Detections were strongly correlated with high IR excess and high inclination, although polarimetry outperformed total intensity angular differential imaging for detecting low inclination disks (${lesssim} 70 deg$). Based on post-survey statistics, we improved upon our pre-survey target prioritization metric predicting polarimetric disk detectability. We also examined scattered-light disks in the contexts of gas, far-IR, and millimeter detections. Comparing $H$-band and ALMA fluxes for two disks revealed tentative evidence for differing grain properties. Finally, we found no preference for debris disks to be detected in scattered light if wide-separation substellar companions were present.
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