The ACS Virgo Cluster Survey consists of HST-ACS Advance Camera for Surveys) imaging for 100 early-type galaxies in the Virgo Cluster, observed in the F475W (~SDSS g) and F850LP (~SDSS z) filters. We derive distances for 84 of these galaxies using the method of surface brightness fluctuations (SBFs), present the SBF distance catalog, and use this database to examine the three-dimensional distribution of early-type galaxies in the Virgo Cluster. The SBF distance moduli have a mean (random) measurement error of 0.07mag (0.5Mpc), or roughly 3 times better than previous SBF measurements for Virgo Cluster galaxies. Five galaxies lie at a distance of d~23Mpc and are members of the W' cloud. The remaining 79 galaxies have a narrow distribution around our adopted distance of <d>=16.5+/-0.1 (random mean error) +/-1.1Mpc (systematic). The rms distance scatter of this sample is {sigma}(d)=0.6+/-0.1Mpc, with little or no dependence on morphological type or luminosity class (i.e., 0.7+/-0.1 and 0.5+/-0.1Mpc for the giants and dwarfs, respectively). The back-to-front depth of the cluster measured from our sample of early-type galaxies is 2.4+/-0.4Mpc (i.e., +/-2{sigma} of the intrinsic distance distribution). The M87 (cluster A) and M49 (cluster B) subclusters are found to lie at distances of 16.7+/-0.2 and 16.4+/-0.2Mpc, respectively. There may be a third subcluster associated with M86. A weak correlation between velocity and line-of-sight distance may be a faint echo of the cluster velocity distribution not having yet completely virialized. In three dimensions, Virgo's early-type galaxies appear to define a slightly triaxial distribution, with axis ratios of (1:0.7:0.5). The principal axis of the best-fit ellipsoid is inclined ~20{deg}-40{deg} from the line of sight, while the galaxies belonging to the W' cloud lie on an axis inclined by ~10{deg}-15{deg}.