  We present rest-frame optical spectra of 60 faint ($R_{AB}\sim 27$; $L\sim0.1 L_*$) Ly$\alpha$-selected galaxies (LAEs) at $z\approx2.56$. The average LAE is consistent with the extreme low-metallicity end of the continuum-selected galaxy distribution at $z\approx2-3$. In particular, the LAEs have extremely high [OIII] $\lambda$5008/H$\beta$ ratios (log([OIII]/H$\beta$) $\sim$ 0.8) and low [NII] $\lambda$6585/H$\alpha$ ratios (log([NII]/H$\alpha$) $<-1.15$). Using the [OIII] $\lambda$4364 auroral line, we find that the star-forming regions in faint LAEs are characterized by high electron temperatures ($T_e\approx1.8\times10^4$K), low oxygen abundances (12 + log(O/H) $\approx$ 8.04, $Z_{neb}\approx0.22Z_\odot$), and high excitations with respect to more luminous galaxies. Our faintest LAEs have line ratios consistent with even lower metallicities, including six with 12 + log(O/H) $\approx$ 6.9$-$7.4 ($Z_{neb}\approx0.02-0.05Z_\odot$). We interpret these observations in light of new models of stellar evolution (including binary interactions). We find that strong, hard ionizing continua are required to reproduce our observed line ratios, suggesting that faint galaxies are efficient producers of ionizing photons and important analogs of reionization-era galaxies. Furthermore, we investigate physical trends accompanying Ly$\alpha$ emission across the largest current sample of combined Ly$\alpha$ and rest-optical galaxy spectroscopy, including 60 faint LAEs and 368 more luminous galaxies at similar redshifts. We find that Ly$\alpha$ emission is strongly correlated with nebular excitation and ionization and weakly correlated with dust attenuation, suggesting that metallicity plays a strong role in determining the observed properties of these galaxies by modulating their stellar spectra, nebular excitation, and dust content. 
The Rest-Frame Optical Spectroscopic Properties of Ly$\alpha$-Emitters   at $z\sim2.5$: The Physical Origins of Strong Ly$\alpha$ Emission