We analyze X-ray spatial and spectral data on the giant elliptical galaxy NGC 4472, the brightest galaxy in the Virgo cluster. The X-ray contours of NGC 4472 are elongated in the northeast-southwest direction, perhaps as a result of motion through the Virgo intracluster gas. A bow shock-like structure is evident on the galaxy's north side. The temperature at a given radius in this bow shock region is slightly higher than the temperature at the same radius on the galaxy's southwest side. Away from this bow shock region, the surface brightness profile of NGC 4472 can be traced out to a radius of 260 kpc in the southwest direction. Beyond 260 kpc, we find evidence for emission from both the Virgo cluster and the Galactic North Polar Spur (believed to be the rim of a hot Galactic superbubble). NGC 4472 is interacting with the dwarf irregular galaxy UGC 7636. We do not detect any excess or deficit in the X-ray emission toward this galaxy. An H I cloud, detected previously in the 21 cm line and located midway between the two galaxies, appears to have been removed from the irregular galaxy through either tidal interaction or ram pressure stripping. We find a marginally significant hole in the ROSAT HRI and PSPC X-ray images at the position of this cloud, suggesting that the cloud lies at the front side of NGC 4472. If the hole in the X-ray images is due to soft X-ray absorption, the total gaseous mass of the cloud must be at least 1.7 x 10(9) M., far greater than its 21 cm H I mass. This suggests that the majority of the material in the cloud is molecular.