  We investigate the connection between globular clusters and ultra-compact
dwarf galaxies (UCDs) by examining the properties of several compact objects
associated with M87, all of which were previously classified as globular
clusters. Combining imaging from the Hubble Space Telescope with ground-based
Keck spectroscopy, we find two objects to have half-light radii, velocity
dispersions and mass-to-light ratios that are consistent with the predictions
of population synthesis models for old, metal-rich, luminous globular clusters.
Three other objects are much larger, with half-light radii of approximately
20pc, and have V-band mass-to-light ratios in the range 6-9. These objects,
which we consider to be UCDs, resemble the nuclei of nucleated dwarf elliptical
galaxies in Virgo, having similar mass-to-light ratios, luminosities and
colors. These UCDs are found to obey the extrapolated scaling relations of
galaxies more closely than those of Galactic globular clusters. There appears
to be a transition between the two types of stellar systems at a mass of about
two million solar masses. If the UCDs are gravitationally bound, then we
suggest that the presence of dark matter is the fundamental property
distinguishing globular clusters from UCDs. More than half of the UCD
candidates uncovered in the ACS Virgo Cluster Survey are associated with a
single galaxy -- M87 -- which suggests that proximity to the Virgo center may
be of critical importance for the formation of these objects. These results
show that distinguishing bonafide UCDs from bright globular clusters requires a
careful analysis of their detailed structural and dynamical properties,
particularly their mass-to-light ratios. (ABRIDGED)
