On 17 August 2017, the LIGO and Virgo observatories made the first direct detection of gravitational waves from the coalescence of a neutron star binary system. The detection of this gravitational-wave signal, GW170817, offers a novel opportunity to directly probe the properties of matter at the extreme conditions found in the interior of these stars. The initial, minimal-assumption analysis of the LIGO and Virgo data placed constraints on the tidal effects of the coalescing bodies, which were then translated to constraints on neutron star radii. Here, we expand upon previous analyses by working under the hypothesis that both bodies were neutron stars that are described by the same equation of state and have spins within the range observed in Galactic binary neutron stars. Our analysis employs two methods: the use of equation-of-state-insensitive relations between various macroscopic properties of the neutron stars and the use of an efficient parametrization of the defining function p(ρ) of the equation of state itself. From the LIGO and Virgo data alone and the first method, we measure the two neutron star radii as R_1=10.8_-1.7{\^}+2.0  km for the heavier star and R_2=10.7_-1.5{\^}+2.1  km for the lighter star at the 90\% credible level. If we additionally require that the equation of state supports neutron stars with masses larger than 1.97  M_⊙ as required from electromagnetic observations and employ the equation-of-state parametrization, we further constrain R_1=11.9_-1.4{\^}+1.4  km and R_2=11.9_-1.4{\^}+1.4  km at the 90\% credible level. Finally, we obtain constraints on p(ρ) at supranuclear densities, with pressure at twice nuclear saturation density measured at 3.5_-1.7{\^}+2.7×10{\^}34  dyn cm{\^}-2 at the 90\% level.