Using deep, high resolution optical imaging from the Next Generation Virgo Cluster Survey we study the properties of nuclear star clusters (NSCs) in a sample of nearly 400 quiescent galaxies in the core of Virgo with stellar masses $10^{5}lesssim M_{*}/M_{odot} lesssim10^{12}$. The nucleation fraction reaches a peak value $f_{n}approx90%$ for $M_{*} approx 10^{9} M_{odot}$ galaxies and declines for both higher and lower masses, but nuclei populate galaxies as small as $M_{*} approx5times10^{5} M_{odot}$. Comparison with literature data for nearby groups and clusters shows that at the low-mass end nucleation is more frequent in denser environments. The NSC mass function peaks at $M_{NSC}approx7times10^{5} M_{odot}$, a factor 3-4 times larger than the turnover mass for globular clusters (GCs). We find a nonlinear relation between the stellar masses of NSCs and of their host galaxies, with a mean nucleus-to-galaxy mass ratio that drops to $M_{NSC}/M_{*}approx3.6times10^{-3}$ for $M_{*} approx 5times10^{9} M_{odot}$ galaxies. Nuclei in both more and less massive galaxies are much more prominent: $M_{NSC}propto M_{*}^{0.46}$ at the low-mass end, where nuclei are nearly 50% as massive as their hosts. We measure an intrinsic scatter in NSC masses at fixed galaxy stellar mass of 0.4 dex, which we interpret as evidence that the process of NSC growth is significantly stochastic. At low galaxy masses we find a close connection between NSCs and GC systems, including a very similar occupation distribution and comparable total masses. We discuss these results in the context of current dissipative and dissipationless models of NSC formation.
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