  The precise localization of the repeating fast radio burst (FRB 121102) has
provided the first unambiguous association (chance coincidence probability
$p\lesssim3\times10^{-4}$) of an FRB with an optical and persistent radio
counterpart. We report on optical imaging and spectroscopy of the counterpart
and find that it is an extended ($0.6^{\prime\prime}-0.8^{\prime\prime}$)
object displaying prominent Balmer and [OIII] emission lines. Based on the
spectrum and emission line ratios, we classify the counterpart as a
low-metallicity, star-forming, $m_{r^\prime} = 25.1$ AB mag dwarf galaxy at a
redshift of $z=0.19273(8)$, corresponding to a luminosity distance of 972 Mpc.
From the angular size, the redshift, and luminosity, we estimate the host
galaxy to have a diameter $\lesssim4$ kpc and a stellar mass of
$M_*\sim4-7\times 10^{7}\,M_\odot$, assuming a mass-to-light ratio between 2 to
3$\,M_\odot\,L_\odot^{-1}$. Based on the H$\alpha$ flux, we estimate the star
formation rate of the host to be $0.4\,M_\odot\,\mathrm{yr^{-1}}$ and a
substantial host dispersion measure depth $\lesssim 324\,\mathrm{pc\,cm^{-3}}$.
The net dispersion measure contribution of the host galaxy to FRB 121102 is
likely to be lower than this value depending on geometrical factors. We show
that the persistent radio source at FRB 121102's location reported by Marcote
et al (2017) is offset from the galaxy's center of light by $\sim$200 mas and
the host galaxy does not show optical signatures for AGN activity. If FRB
121102 is typical of the wider FRB population and if future interferometric
localizations preferentially find them in dwarf galaxies with low metallicities
and prominent emission lines, they would share such a preference with long
gamma ray bursts and superluminous supernovae.
