  We present a study of ultra-diffuse galaxies (UDGs) in the Virgo Cluster
based on deep imaging from the Next Generation Virgo Cluster Survey (NGVS).
Applying a new definition for the UDG class based on galaxy scaling relations,
we define samples of 44 and 26 UDGs using expansive and restrictive selection
criteria, respectively. Our UDG sample includes objects that are significantly
fainter than previously known UDGs: i.e., more than half are fainter than
$\langle\mu\rangle_e \sim27.5$ mag arcsec$^{-2}$. The UDGs in Virgo's core
region show some evidence for being structurally distinct from "normal" dwarf
galaxies, but this separation disappears when considering the full sample of
galaxies throughout the cluster. UDGs are more centrally concentrated in their
spatial distribution than other Virgo galaxies of similar luminosity, while
their morphologies demonstrate that at least some UDGs owe their diffuse nature
to physical processes---such as tidal interactions or low-mass mergers---that
are at play within the cluster environment. The globular cluster (GC) systems
of Virgo UDGs have a wide range in specific frequency ($S_N$), with a higher
mean $S_N$ than "normal" Virgo dwarfs, but a lower mean $S_N$ than Coma UDGs at
fixed luminosity. Their GCs are predominantly blue, with a small contribution
from red clusters in the more massive UDGs. The combined GC luminosity function
is consistent with those observed in dwarf galaxies, showing no evidence of
being anomalously luminous. The diversity in their morphologies and their GC
properties suggests no single process has given rise to all objects within the
UDG class. Based on the available evidence, we conclude that UDGs are simply
those systems that occupy the extended tails of the galaxy size and surface
brightness distributions.
