We present results from a 38 ks Chandra X-ray observation of the z = 0.059 galaxy cluster A1991. The cluster has a bright X-ray core and a central temperature gradient that declines inward from 2.7 keV at 130 kpc to ≈1.6 keV at the cluster center. The radiative cooling time of the gas in the inner 10 kpc is about 0.5 Gyr, and it rises to 1 Gyr at a radius of 20 kpc. The cooling rate of the gas within the latter radius is ≲25 M⊙ yr-1. The Chandra ACIS-S3 image shows that the intracluster medium has an asymmetric surface brightness distribution with respect to the central galaxy. Bright knots of soft X-ray emission embedded in a cometary structure are located approximately 10&arcsec; north of the optical center of the cD galaxy. Unlike the structures seen in other cooling flow clusters, the knots have no obvious association with the radio source. The structure's temperature of 0.83 keV makes it nearly 1 keV cooler than its surroundings, and its mass is 3.4 × 109 M⊙. Based on its bow-shaped appearance and modest overpressure with respect to its surroundings, we interpret the structure as a cool mass concentration that is breaking apart as it travels northward through the center of the cluster.