  The [NII] 122 and 205 \mu m transitions are powerful tracers of the ionized
gas in the interstellar medium. By combining data from 21 galaxies selected
from the Herschel KINGFISH and Beyond the Peak surveys, we have compiled 141
spatially resolved regions with a typical size of ~1 kiloparsec, with
observations of both [NII] far-infrared lines. We measure [NII] 122/205 line
ratios in the ~0.6-6 range, which corresponds to electron gas densities
$n_e$~1-300 cm$^{-3}$, with a median value of $n_e$=30 cm$^{-3}$. Variations in
the electron density within individual galaxies can be as a high as a factor of
~50, frequently with strong radial gradients. We find that $n_e$ increases as a
function of infrared color, dust-weighted mean starlight intensity, and star
formation rate surface density ($\Sigma_{SFR}$). As the intensity of the [NII]
transitions is related to the ionizing photon flux, we investigate their
reliability as tracers of the star formation rate (SFR). We derive relations
between the [NII] emission and SFR in the low-density limit and in the case of
a log-normal distribution of densities. The scatter in the correlation between
[NII] surface brightness and $\Sigma_{SFR}$ can be understood as a property of
the $n_e$ distribution. For regions with $n_e$ close to or higher than the
[NII] line critical densities, the low-density limit [NII]-based SFR
calibration systematically underestimates the SFR since [NII] emission is
collisionally quenched. Finally, we investigate the relation between [NII]
emission, SFR, and $n_e$ by comparing our observations to predictions from the
MAPPINGS-III code.
