  We present results from near-infrared spectroscopy of 26 emission-line
galaxies at z ~ 2 obtained with the FIRE spectrometer on the Magellan Baade
telescope. The sample was selected from the WISP survey, which uses the
near-infrared grism of the Hubble Space Telescope Wide Field Camera 3 to detect
emission-line galaxies over 0.3 < z < 2.3. Our FIRE follow-up spectroscopy
(R~5000) over 1.0-2.5 micron permits detailed measurements of physical
properties of the z~2 emission-line galaxies. Dust-corrected star formation
rates for the sample range from ~5-100 M_sun yr-1. We derive a median
metallicity for the sample of ~0.45 Z_sun, and the estimated stellar masses
range from ~10^8.5 - 10^9.5 M_sun. The average ionization parameters measured
for the sample are typically much higher than what is found for local
star-forming galaxies. We derive composite spectra from the FIRE sample, from
which we infer typical nebular electron densities of ~100-400 cm^-3. Based on
the location of the galaxies and composite spectra on BPT diagrams, we do not
find evidence for significant AGN activity in the sample. Most of the galaxies
as well as the composites are offset in the BPT diagram toward higher [O
III]/H-beta at a given [N II]/H-alpha, in agreement with other observations of
z > 1 star-forming galaxies, but composite spectra derived from the sample do
not show an appreciable offset from the local star-forming sequence on the [O
III]/H-beta versus [S II]/H-alpha diagram. We infer a high nitrogen-to-oxygen
abundance ratio from the composite spectrum, which may contribute to the offset
of the high-redshift galaxies from the local star-forming sequence in the [O
III]/H-beta versus [N II]/H-alpha diagram. We speculate that the elevated
nitrogen abundance could result from substantial numbers of Wolf-Rayet stars in
starbursting galaxies at z~2. (Abridged)
