We present a comprehensive analysis of 21 light curves of Type 1 active galactic nuclei (AGN) from the Kepler spacecraft. First, we describe the necessity and development of a customized pipeline for treating Kepler data of stochastically variable sources like AGN. We then present the light curves, power spectral density functions (PSDs), and flux histograms. The light curves display an astonishing variety of behaviors, many of which would not be detected in ground-based studies, including switching between distinct flux levels. Six objects exhibit PSD flattening at characteristic timescales that roughly correlate with black hole mass. These timescales are consistent with orbital timescales or free-fall accretion timescales. We check for correlations of variability and high-frequency PSD slope with accretion rate, black hole mass, redshift, and luminosity. We find that bolometric luminosity is anticorrelated with both variability and steepness of the PSD slope. We do not find evidence of the linear rms-flux relationships or lognormal flux distributions found in X-ray AGN light curves, indicating that reprocessing is not a significant contributor to optical variability at the 0.1%-10% level.