  [Abridged] We present a catalog of 68 galaxy clusters, of which 19 are new
discoveries, detected via the Sunyaev-Zel'dovich effect (SZ) at 148 GHz in the
Atacama Cosmology Telescope (ACT) survey of 504 square degrees on the celestial
equator. A subsample of 48 clusters within the 270 square degree region
overlapping SDSS Stripe 82 is estimated to be 90% complete for M_500c > 4.5e14
Msun and 0.15 < z < 0.8. While matched filters are used to detect the clusters,
the sample is studied further through a "Profile Based Amplitude Analysis"
using a single filter at a fixed \theta_500 = 5.9' angular scale. This new
approach takes advantage of the "Universal Pressure Profile" (UPP) to fix the
relationship between the cluster characteristic size (R_500) and the integrated
Compton parameter (Y_500). The UPP scalings are found to be nearly identical to
an adiabatic model, while a model incorporating non-thermal pressure better
matches dynamical mass measurements and masses from the South Pole Telescope. A
high signal to noise ratio subsample of 15 ACT clusters is used to obtain
cosmological constraints. We first confirm that constraints from SZ data are
limited by uncertainty in the scaling relation parameters rather than sample
size or measurement uncertainty. We next add in seven clusters from the ACT
Southern survey, including their dynamical mass measurements based on galaxy
velocity dispersions. In combination with WMAP7 these data simultaneously
constrain the scaling relation and cosmological parameters, yielding \sigma_8 =
0.829 \pm 0.024 and \Omega_m = 0.292 \pm 0.025. The results include
marginalization over a 15% bias in dynamical mass relative to the true halo
mass. In an extension to LCDM that incorporates non-zero neutrino mass density,
we combine our data with WMAP7+BAO+Hubble constant measurements to constrain
\Sigma m_\nu < 0.29 eV (95% C. L.).
