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 $M_*∼4-7\times 10^{7}\,M_⊙$, assuming a mass-to-light ratio between 2 to 3$\,M_⊙\,L_⊙^{-1}$. Based on the H$α$ flux, we estimate the star formation rate of the host to be $0.4\,M_⊙\,\mathrm{yr^{-1}}$ 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 $∼$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. ; Comment: 12 pages, 3 figures, Published in ApJ Letters. V2: Corrected mistake in author list