We present Chandra X-ray observations of the Hydra A cluster of galaxies, and we report the discovery of structure in the central 80 kpc of the cluster's X-ray-emitting gas. The most remarkable structures are depressions in the X-ray surface brightness, ~25-35 kpc in diameter, that are coincident with Hydra A's radio lobes. The depressions are nearly devoid of X-ray-emitting gas, and there is no evidence for shock-heated gas surrounding the radio lobes. We suggest that the gas within the surface brightness depressions was displaced as the radio lobes expanded subsonically, leaving cavities in the hot atmosphere. The gas temperature declines from 4 keV at 70 kpc to 3 keV in the inner 20 kpc of the brightest cluster galaxy (BCG), and the cooling time of the gas is ~600 Myr in the inner 10 kpc. These properties are consistent with the presence of an ~34 M☉ yr-1 cooling flow within a 70 kpc radius. Bright X-ray emission is present in the BCG surrounding a recently accreted disk of nebular emission and young stars. The star formation rate is commensurate with the cooling rate of the hot gas within the volume of the disk, although the sink for the material that may be cooling at larger radii remains elusive. A bright, unresolved X-ray source is present in the BCG's nucleus, coincident with the radio core. Its X-ray spectrum is consistent with a power law absorbed by a foreground NH 4 × 1022 cm-2 column of hydrogen. This column is roughly consistent with the hydrogen column seen in absorption toward the 24 pc diameter VLBA radio source. Apart from the point source, no evidence for excess X-ray absorption above the Galactic column is found.