  [Abridged] We present new measurements of the H-alpha luminosity function
(LF) and SFR volume density for galaxies at z~0.8. Our analysis is based on
1.18$\mu$m narrowband data from the NEWFIRM H-alpha Survey, a comprehensive
program designed to capture deep samples of intermediate redshift emission-line
galaxies using narrowband imaging in the near-infrared. The combination of
depth ($\approx1.9\times10^{-17}$ erg s$^{-1}$ cm$^{-2}$ in H-alpha at
3$\sigma$) and areal coverage (0.82 deg$^2$) complements other recent H-alpha
studies at similar redshifts, and enables us to minimize the impact of cosmic
variance and place robust constraints on the shape of the LF. The present
sample contains 818 NB118 excess objects, 394 of which are selected as H-alpha
emitters. Optical spectroscopy has been obtained for 62% of the NB118 excess
objects. Empirical optical broadband color classification is used to sort the
remainder of the sample. A comparison of the LFs constructed for the four
individual fields reveals significant cosmic variance, emphasizing that
multiple, widely separated observations are required. The dust-corrected LF is
well-described by a Schechter function with L*=10^{43.00\pm0.52} ergs s^{-1},
\phi*=10^{-3.20\pm0.54} Mpc^{-3}, and \alpha=-1.6\pm0.19. We compare our
H-alpha LF and SFR density to those at z<1, and find a rise in the SFR density
\propto(1+z)^{3.4}, which we attribute to significant L* evolution. Our H-alpha
SFR density of 10^{-1.00\pm0.18} M_sun yr^{-1} Mpc^{-3} is consistent with UV
and [O II] measurements at z~1. We discuss how these results compare to other
H-alpha surveys at z~0.8, and find that the different methods used to determine
survey completeness can lead to inconsistent results. This suggests that future
surveys probing fainter luminosities are needed, and more rigorous methods of
estimating the completeness should be adopted as standard procedure.
