We present optical spectra obtained on 1993 April 7 and April 13 of the bright supernova SN 1993J in the nearby galaxy M81. The spectra show strong hydrogen Balmer lines with pronounced P Cygni profiles, except for Halpha, which is greatly dominated by emission. In order to obtain diagnostics on the velocities, temperatures, densities, and composition of the atmosphere, we present non-local thermodynamic equilibrium (NLTE) synthetic spectrum fits to these observations. For April 7 we find a photospheric velocity of 10,000-11,000 km s-1, with a photospheric electron temperature of 7500-8000 K and a radius of 10(15) cm. On April 13 the photospheric velocity is about 11,000 km s-1, the photospheric temperature is about 6500 K, and the radius is almost-equal-to 1.6 x 10(15) cm. Most of the features of the spectra are well fitted by model atmospheres with solar composition. We find a distance for SN 1993J of 4.0 +/- 0.5 Mpc and the explosion epoch to be UT 1993 March 27.5 +/- 0.5.