  (abridged) We present cosmological constraints obtained from galaxy clusters
identified by their Sunyaev-Zel'dovich effect signature in the 2500 square
degree South Pole Telescope Sunyaev Zel'dovich survey. We consider the 377
cluster candidates identified at z>0.25 with a detection significance greater
than five, corresponding to the 95% purity threshold for the survey. We compute
constraints on cosmological models using the measured cluster abundance as a
function of mass and redshift. We include additional constraints from
multi-wavelength observations, including Chandra X-ray data for 82 clusters and
a weak lensing-based prior on the normalization of the mass-observable scaling
relations. Assuming a LCDM cosmology, where the species-summed neutrino mass
has the minimum allowed value (mnu = 0.06 eV) from neutrino oscillation
experiments, we combine the cluster data with a prior on H0 and find sigma_8 =
0.797+-0.031 and Omega_m = 0.289+-0.042, with the parameter combination
sigma_8(Omega_m/0.27)^0.3 = 0.784+-0.039. These results are in good agreement
with constraints from the CMB from SPT, WMAP, and Planck, as well as with
constraints from other cluster datasets. Adding mnu as a free parameter, we
find mnu = 0.14+-0.08 eV when combining the SPT cluster data with Planck CMB
data and BAO data, consistent with the minimum allowed value. Finally, we
consider a cosmology where mnu and N_eff are fixed to the LCDM values, but the
dark energy equation of state parameter w is free. Using the SPT cluster data
in combination with an H0 prior, we measure w = -1.28+-0.31, a constraint
consistent with the LCDM cosmological model and derived from the combination of
growth of structure and geometry. When combined with primarily geometrical
constraints from Planck CMB, H0, BAO and SNe, adding the SPT cluster data
improves the w constraint from the geometrical data alone by 14%, to w =
-1.023+-0.042.
