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We present new imaging and spectroscopic observations of six ultracompact dwarf (UCD) galaxies in the Virgo Cluster, along with reanalyzed data for five Fornax Cluster UCDs. These are the most luminous UCDs: -14 mag<M_V←12 mag. Our Hubble Space Telescope imaging shows that most of the UCDs have shallow or steep cusps in their cores; only one UCD has a flat ``King'' core. None of the UCDs show tidal cutoffs down to our limiting surface brightness. Spectroscopic analysis shows that Virgo UCDs are old (older than 8 Gyr) and have metallicities in the range from [Z/H]=-1.35 to +0.35 dex. Five Virgo UCDs have supersolar [α/Fe] abundance ratios, and one Virgo UCD has a solar abundance ratio. The supersolar [α/Fe] abundances are typical of old stellar populations found in globular clusters and elliptical galaxies. We find that Virgo UCDs have structural and dynamical properties similar to Fornax UCDs. The Virgo and Fornax UCDs all have masses ~(2-9)x10⁷ M_☉ and mass-to-light ratios ~(3-5) M_☉/L_☉,V. The dynamical mass-to-light ratios for Virgo UCDs are consistent with simple stellar population model predictions: the Virgo UCDs do not require dark matter to explain their mass-to-light ratios. We conclude that the internal properties of Virgo UCDs are consistent with them being the high-mass/high-luminosity extreme of known globular cluster populations. We refrain from any firm conclusions on Fornax UCD origins until accurate age, metallicity, and α-abundance estimates are obtained for them. Some of our results, notably the fundamental plane projections, are consistent with the formation of UCDs by the simple removal of the halo from the nuclei of nucleated dwarf galaxies. However, the ages, metallicities, and abundances for Virgo UCDs are not consistent with this simple stripping model. It might be consistent with more sophisticated models of the stripping process that include the effects of gas removal on the chemical evolution of the nuclei.