  We present Keck/NIRSPEC spectroscopic observations of three Lyman alpha
emitting galaxies (LAEs) at z ~ 2.3 discovered with the HETDEX pilot survey. We
detect Halpha, [OIII], and Hbeta emission from two galaxies at z = 2.29 and
2.49, designated HPS194 and HPS256, respectively, representing the first
detection of multiple rest-frame optical emission lines in galaxies at
high-redshift selected on the basis of their Lyman alpha emission. The
redshifts of the Lyman alpha emission from these galaxies are offset redward of
the systemic redshifts by Delta_v = 162 +/- 37 (photometric) +/- 42
(systematic) km/s for HPS194, and Delta_v = 36 +/- 35 +/- 18 km/s for HPS256.
An interpretation for HPS194 is that a large-scale outflow may be occurring in
its interstellar medium. The emission line ratios imply that neither LAE hosts
an active galactic nucleus. Using the upper limits on the [NII] emission we
place meaningful constraints on the gas-phase metallicities in these two LAEs
of Z < 0.17 and < 0.28 Zsol (1 sigma). Measuring the stellar masses of these
objects via spectral energy distribution (SED) fitting (~ 10^10 and 6 x 10^8
Msol, respectively), we study the nature of LAEs in a mass-metallicity plane.
At least one of these two LAEs appears to be more metal poor than
continuum-selected star-forming galaxies at the same redshift and stellar mass,
implying that objects exhibiting Lyman alpha emission may be systematically
less chemically enriched than the general galaxy population. We use the SEDs of
these two galaxies to show that neglecting the emission lines when fitting
stellar population models to the observed photometry can result in
overestimates of the population age by orders of magnitude, and the stellar
mass by a factor of ~ 2. This effect is particularly important at z > 7, where
similarly strong emission lines may masquerade in the photometry as a 4000 A
break (abridged).
