We present a catalog of 23,790 extended low-surface-brightness galaxies (LSBGs) identified in $sim 5000 deg^2$ from the first three years of imaging data from the Dark Energy Survey (DES). Based on a single-component Sersic model fit, we define extended LSBGs as galaxies with $g$-band effective radii $R_{eff}(g) > 2.5$ and mean surface brightness $bar{mu}_{eff}(g) > 24.2 ,mag .arcsec^{-2}$. We find that the distribution of LSBGs is strongly bimodal in $(g-r)$ vs. $(g-i$) color space. We divide our sample into red ($g-i geq 0.60$) 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 $sim 33%$ 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 41 objects that can be classified as ultra-diffuse galaxies, defined as LSBGs with projected physical effective radii $R_{eff} > 1.5 ,kpc$ and central surface brighthness $mu_0(g) > 24.0, mag ,arcsec^{-2}$. The wide-area sample of LSBGs in DES can be used to test the role of environment on models of LSBG formation and evolution.
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