The Next Generation Virgo Cluster Survey. XXXIV. Ultra-Compact Dwarf (UCD) Galaxies in the Virgo Cluster

70 pages, 24 figures, 6 tables, accepted for publication in ApJSInternational audienceWe present a study of ultra compact dwarf (UCD) galaxies in the Virgo cluster based mainly on imaging from the Next Generation Virgo Cluster Survey (NGVS). Using $\sim$100 deg$^{2}$ of $u^*giz$ imaging, we have identified more than 600 candidate UCDs, from the core of Virgo out to its virial radius. Candidates have been selected through a combination of magnitudes, ellipticities, colors, surface brightnesses, half-light radii and, when available, radial velocities. Candidates were also visually validated from deep NGVS images. Subsamples of varying completeness and purity have been defined to explore the properties of UCDs and compare to those of globular clusters and the nuclei of dwarf galaxies with the aim of delineating the nature and origins of UCDs. From a surface density map, we find the UCDs to be mostly concentrated within Virgo's main subclusters, around its brightest galaxies. We identify several subsamples of UCDs -- i.e., the brightest, largest, and those with the most pronounced and/or asymmetric envelopes -- that could hold clues to the origin of UCDs and possible evolutionary links with dwarf nuclei. We find some evidence for such a connection from the existence of diffuse envelopes around some UCDs, and comparisons of radial distributions of UCDs and nucleated galaxies within the cluster