We use HST/ACS imaging of 100 early-type galaxies in the ACS Virgo Cluster Survey to investigate the nature of diffuse star clusters (DSCs). Compared to globular clusters (GCs), these star clusters have moderately low luminosities (M_V > -8) and a broad distribution of sizes (3 < r_h < 30 pc), but they are principally characterized by their low mean surface brightnesses which can be more than three magnitudes fainter than a typical GC (mu_g > 20 mag arcsec^-2). The median colors of diffuse star cluster systems are red, 1.1 < g-z < 1.6, which is redder than metal-rich GCs and often as red as the galaxy itself. Most DSC systems thus have mean ages older than 5 Gyr or else have super-solar metallicities. We find that 12 galaxies in our sample contain a significant excess of diffuse star cluster candidates -- nine are lenticulars (S0s), and five visibly contain dust. We also find a substantial population of DSCs in the halo of the giant elliptical M49, associated with the companion galaxy VCC 1199. Most DSC systems appear to be both aligned with the galaxy light and associated with galactic disks, but at the same time many lenticulars do not host substantial DSC populations. Diffuse star clusters in our sample share similar characteristics to those identified in other nearby lenticular, spiral, and dwarf galaxies, and we suggest that DSCs preferentially form, survive, and coevolve with galactic disks. Their properties are broadly consistent with those of merged star cluster complexes, and we note that despite being 3 - 5 magnitudes brighter than DSCs, ultra-compact dwarfs have similar surface brightnesses. The closest Galactic analogs to the DSCs are the old open clusters. We suggest that if a diffuse star cluster population did exist in the disk of the Milky Way, it would be very difficult to find. (Abridged)
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