We present a catalog of 23,790 extended low-surface-brightness galaxies (LSBGs) identified in ~5000 deg2 from the first three years of imaging data from the Dark Energy Survey (DES). Based on a single-component Sérsic model fit, we define extended LSBGs as galaxies with g-band effective radii Reff (g) > 2."5 and mean surface brightness μmeff (g) > 24.2 mag arcsec-2. We find that the distribution of LSBGs is strongly bimodal in (g - r) versus (g - i) color space. We divide our sample into red (g - i ≥ 0.60) and blue (g - i < 0.60) galaxies and study the properties of the two populations. Redder LSBGs are more clustered than their blue counterparts and are correlated with the distribution of nearby (z < 0.10) bright galaxies. Red LSBGs constitute ~33% of our LSBG sample, and ~30% of these are located within 1° of low-redshift galaxy groups and clusters (compared to ~8% of the blue LSBGs). For nine of the most prominent galaxy groups and clusters, we calculate the physical properties of associated LSBGs assuming a redshift derived from the host system. In these systems, we identify 41 objects that can be classified as ultradiffuse galaxies, defined as LSBGs with projected physical effective radii Reff > 1.5 kpc and central surface brightness μ0 (g) > 24.0 mag arcsec-2. The wide-area sample of LSBGs in DES can be used to test the role of environment on models of LSBG formation and evolution.