  We present optical spectroscopy of two samples of GALEX grism selected Ly
alpha emitters (LAEs): one at z=0.195-0.44 and the other at z=0.65-1.25. We
have also observed a comparison sample of galaxies in the same redshift
intervals with the same UV magnitude distributions but with no detected Ly
alpha. We use the optical spectroscopy to eliminate active galactic nuclei
(AGNs) and to obtain the optical emission-line properties of the samples. We
compare the luminosities of the LAEs in the two redshift intervals and show
that there is dramatic evolution in the maximum Ly alpha luminosity over z=0-1.
Focusing on the z=0.195-0.44 samples alone, we show that there are tightly
defined relations between all of the galaxy parameters and the rest-frame
equivalent width (EW) of H alpha. The higher EW(H alpha) sources all have lower
metallicities, bluer colors, smaller sizes, and less extinction, consistent
with their being in the early stages of the galaxy formation process. We find
that 75 +- 12% of the LAEs have EW(H alpha)>100 Angstrom, and, conversely, that
31 +/- 3% of galaxies with EW(H alpha)>100 Angstrom are LAEs. We correct the
broadband magnitudes for the emission-line contributions and use spectral
synthesis fits to estimate the ages of the galaxies. We find a median age of
1.1x10^{8} yr for the LAE sample and 1.4x10^{9} yr for the UV-continuum sample
without detected Ly alpha. The median metallicity of the LAE sample is
12+log(O/H)=8.24, or about 0.4 dex lower than the UV-continuum sample.
