(Abridged) The ACS Virgo Cluster Survey is an HST program to obtain high-resolution, g and z-band images for 100 early-type members of the Virgo Cluster, spanning a range of ~460 in blue luminosity. Based on this large, homogeneous dataset, we present a sharp upward revision in the frequency of nucleation in early-type galaxies brighter than M_B ~ -15 (66 < f_n < 82%), and find no evidence for nucleated dwarfs to be more concentrated to the center of Virgo than their non-nucleated counterparts. Resolved stellar nuclei are not present in galaxies brighter than M_B ~ -20.5, however, there is no clear evidence from the properties of the nuclei, or from the overall incidence of nucleation, for a change at M_B ~ -17.6, the traditional dividing point between dwarf and giant galaxies. On average, nuclei are ~3.5 mag brighter than a typical globular cluster and have a median half-light radius ~4.2 pc. Nuclear luminosities correlate with nuclear sizes and, in galaxies fainter than M_B ~ -17.6, nuclear colors. Comparing the nuclei to the nuclear clusters found in late-type spiral galaxies reveals a close match in terms of size, luminosity and overall frequency, pointing to a formation mechanism that is rather insensitive to the detailed properties of the host galaxy. The mean nuclear-to-galaxy luminosity ratio is indistinguishable from the mean SBH-to-bulge mass ratio, calculated in early-type galaxies with detected supermassive black holes (SBHs). We argue that compact stellar nuclei might be the low-mass counterparts of the SBHs detected in the bright galaxies, and that one should think in terms of Central Massive Objects -- either SBHs or compact stellar nuclei -- that accompany the formation of almost all early-type galaxies and contain a mean fraction ~0.3% of the total bulge mass.
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