  Recent studies have shown that the number counts of convergence peaks
N(kappa) in weak lensing (WL) maps, expected from large forthcoming surveys,
can be a useful probe of cosmology. We follow up on this finding, and use a
suite of WL convergence maps, obtained from ray-tracing N-body simulations, to
study (i) the physical origin of WL peaks with different heights, and (ii)
whether the peaks contain information beyond the convergence power spectrum
P_ell. In agreement with earlier work, we find that high peaks (with amplitudes
>~ 3.5 sigma, where sigma is the r.m.s. of the convergence kappa) are typically
dominated by a single massive halo. In contrast, medium-height peaks (~0.5-1.5
sigma) cannot be attributed to a single collapsed dark matter halo, and are
instead created by the projection of multiple (typically, 4-8) halos along the
line of sight, and by random galaxy shape noise. Nevertheless, these peaks
dominate the sensitivity to the cosmological parameters w, sigma_8, and
Omega_m. We find that the peak height distribution and its dependence on
cosmology differ significantly from predictions in a Gaussian random field. We
directly compute the marginalized errors on w, sigma_8, and Omega_m from the
N(kappa) + P_ell combination, including redshift tomography with source
galaxies at z_s=1 and z_s=2. We find that the N(kappa) + P_ell combination has
approximately twice the cosmological sensitivity compared to P_ell alone. These
results demonstrate that N(kappa) contains non-Gaussian information
complementary to the power spectrum.
