  We interpret and model the statistical weak lensing measurements around
130,000 groups and clusters of galaxies in the Sloan Digital Sky Survey
presented by Sheldon et al. 2007 (Paper I). We present non-parametric
inversions of the 2D shear profiles to the mean 3D cluster density and mass
profiles in bins of both optical richness and cluster i-band luminosity. We
correct the inferred 3D profiles for systematic effects, including non-linear
shear and the fact that cluster halos are not all precisely centered on their
brightest galaxies. We also model the measured cluster shear profile as a sum
of contributions from the brightest central galaxy, the cluster dark matter
halo, and neighboring halos. We infer the relations between mean cluster virial
mass and optical richness and luminosity over two orders of magnitude in
cluster mass; the virial mass at fixed richness or luminosity is determined
with a precision of 13% including both statistical and systematic errors. We
also constrain the halo concentration parameter and halo bias as a function of
cluster mass; both are in good agreement with predictions of LCDM models. The
methods employed here will be applicable to deeper, wide-area optical surveys
that aim to constrain the nature of the dark energy, such as the Dark Energy
Survey, the Large Synoptic Survey Telescope and space-based surveys.
