  (abridged) We apply the Jeans equation to estimate masses for eight of the
brightest dSph galaxies. For Fornax we obtain a model-independent constraint on
the maximum-circular velocity, Vmax=18_{-3}^{+5} km/s. Although we obtain only
lower-limits of Vmax > 10 km/s for the remaining dSphs, we find that in all
cases the enclosed mass at the projected half-light radius is well constrained
and robust to a wide range of halo models and velocity anisotropies. We derive
a simple analytic formula that estimates M(rhalf) accurately with respect to
results from the full Jeans analysis. Applying this formula to the entire
population of Local Group dSphs, we demonstrate a correlation such that
M(rhalf)\propto rhalf^{1.4\pm 0.4}, or in terms of the mean density interior to
the half-light radius, <rho> \propto rhalf^{-1.6\pm 0.4}. This relation is
driven by the fact that the dSph data exhibit a correlation between global
velocity dispersion and half-light radius. We argue that tidal forces are
unlikely to have introduced this relation, but tides may have increased the
scatter and/or altered the slope. While the data are well described by mass
profiles ranging over a factor of < 2 in normalization (Vmax ~ 10-20 km/s), we
consider the hypothesis that all dSphs are embedded within a "universal" dark
matter halo. We show that in addition to the power law M\propto r^{1.4}, viable
candidates include a cuspy "NFW" halo with Vmax ~ 15 km/s and scale radius r_0
~ 800 pc, as well as a cored halo with Vmax ~ 13 km/s and r_0 ~ 150 pc.
Finally, assuming that their measured velocity dispersions accurately reflect
their masses, the smallest dSphs now allow us to resolve dSph densities at
radii as small as a few tens of pc. At these small scales we find mean
densities as large as <\rho> ~ 5 Msun/pc^3 (200 GeV/cm^3).
