We present a catalog of 20,977 extended low-surface-brightness galaxies (LSBGs) identified in ~ 5000 deg$^2$ 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 $R_{scriptsize{eff}} > 2.5"$ and mean surface brightness $ar{mu}_{scriptsize{eff}}(g) > 24.3$ mag arcsec$^{-2}$. We find that the distribution of LSBGs is strongly bimodal in $(g-r)$ vs. $(g-i$) color space. We divide our sample into red ($g-i geq 0.59$) and blue ($g-i<0.59$) 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 $sim 35%$ of our LSBG sample, and $sim 30%$ of these are located within 1 deg of low-redshift galaxy groups and clusters (compared to $sim 8%$ 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 108 objects that can be classified as ultra-diffuse galaxies, defined as LSBGs with projected physical effective radii $R_{scriptsize{eff}} > 1.5$ kpc. The wide-area sample of LSBGs in DES can be used to test the role of environment on models of LSBG formation and evolution....
