We present the discovery of the Kepler-20 planetary system, which we initially identified through the detection of five distinct periodic transit signals in the Kepler light curve of the host star 2MASS J19104752+4220194. From high-resolution spectroscopy of the star, we find a stellar effective temperature T_eff_= 5455+/-100 K, a metallicity of [Fe/H] = 0.01+/-0.04, and a surface gravity of log g = 4.4+/-0.1. We combine these estimates with an estimate of the stellar density derived from the transit light curves to deduce a stellar mass of M_*_ = 0.912+/-0.034 M_{sun}_and a stellar radius of R_*_ = 0.944^+0.060^_-0.095_R_{sun}_. For three of the transit signals, we demonstrate that our results strongly disfavor the possibility that these result from astrophysical false positives. We determine the orbital periods and physical radii of the three confirmed planets to be 3.70 days and 1.91^+0.12^_-0.21_R_{earth}_ for Kepler-20b, 10.85 days and 3.07^+0.20^_-0.31_R_{earth}_ for Kepler-20c, and 77.61 days and 2.75^+0.17^_-0.30_R_{earth}_ for Kepler-20d. From multi-epoch radial velocities, we determine the masses of Kepler-20b and Kepler-20c to be 8.7+/-2.2 M_{earth}_ and 16.1 +/-3.5 M_{earth}_, respectively, and we place an upper limit on the mass of Kepler-20d of 20.1 M_{earth}_(2{sigma}).