We present a search for RR Lyrae stars using the full six-year data set from the Dark Energy Survey covering ~5000deg^2^ of the southern sky. Using a multistage multivariate classification and light-curve template-fitting scheme, we identify RR Lyrae candidates with a median of 35 observations per candidate. We detect 6971 RR Lyrae candidates out to ~335kpc, and we estimate that our sample is >70% complete at ~150kpc. We find excellent agreement with other wide-area RR Lyrae catalogs and RR Lyrae studies targeting the Magellanic Clouds and other Milky Way satellite galaxies. We fit the smooth stellar halo density profile using a broken-power-law model with fixed halo flattening (q=0.7), and we find strong evidence for a break at R_0_=32.1_-0.9_^+1.1^kpc with an inner slope of n_1_=-2.54_-0.09_^+0.09^ and an outer slope of n_2_=-5.42_-0.14_^+0.13^. We use our catalog to perform a search for Milky Way satellite galaxies with large sizes and low luminosities. Using a set of simulated satellite galaxies, we find that our RR Lyrae-based search is more sensitive than those using resolved stellar populations in the regime of large (rh>=500pc), low-surface-brightness dwarf galaxies. A blind search for large, diffuse satellites yields three candidate substructures. The first can be confidently associated with the dwarf galaxy Eridanus II. The second has a distance and proper motion similar to the ultrafaint dwarf galaxy Tucana II but is separated by ~5 deg. The third is close in projection to the globular cluster NGC 1851 but is ~10 kpc more distant and appears to differ in proper motion.