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)
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