We report our identification of the optical afterglow and host galaxy of the short-duration gamma-ray burst sGRB 160821B. The spectroscopic redshift of the host is z = 0.162, making it one of the lowest redshift short-duration gamma-ray bursts (sGRBs) identified by Swift. Our intensive follow-up campaign using a range of ground-based facilities as well as Hubble Space Telescope, XMM-Newton, and Swift, shows evidence for a late-time excess of optical and near-infrared emission in addition to a complex afterglow. The afterglow light curve at X-ray frequencies reveals a narrow jet, theta(j) similar to 1.9(-0.03)(+0.10) deg, that is refreshed at >1 day post-burst by a slower outflow with significantly more energy than the initial outflow that produced the main GRB. Observations of the 5 GHz radio afterglow shows a reverse shock into a mildly magnetized shell. The optical and near-infrared excess is fainter than AT2017gfo associated with GW170817, and is well explained by a kilonova with dynamic ejecta mass M-dyn = (1.0 +/- 0.6) x 10(-3) M-circle dot and a secular (post-merger) ejecta mass with M-pm = (1.0 +/- 0.6) x 10(-2) M-circle dot, consistent with a binary neutron star merger resulting in a short-lived massive neutron star. This optical and near-infrared data set provides the best-sampled kilonova light curve without a gravitational wave trigger to date.