We present a resolved star spectroscopic survey of 15 dwarf spheroidal (dSph) satellites of the Andromeda galaxy (M31). We filter foreground contamination from Milky Way (MW) stars, noting that MW substructure is evident in this contaminant sample. We also filter M31 halo field giant stars and identify the remainder as probable dSph members. We then use these members to determine the kinematical properties of the dSphs. For the first time, we confirm that And XVIII, XXI, and XXII show kinematics consistent with bound, dark-matter-dominated galaxies. From the velocity dispersions for the full sample of dSphs we determine masses, which we combine with the size and luminosity of the galaxies to produce mass-size-luminosity scaling relations. With these scalings we determine that the M31 dSphs are fully consistent with the MW dSphs, suggesting that the well-studied MW satellite population provides a fair sample for broader conclusions. We also estimate dark matter halo masses of the satellites and find that there is no sign that the luminosity of these galaxies depends on their dark halo mass, a result consistent with what is seen for MW dwarfs. Two of the M31 dSphs (And XV, XVI) have estimated maximum circular velocities smaller than 12 km/s (to 1{sigma}), which likely places them within the lowest-mass dark matter halos known to host stars (along with Bootes I of the MW). Finally, we use the systemic velocities of the M31 satellites to estimate the mass of the M31 halo, obtaining a virial mass consistent with previous results.