Pulsar timing arrays are galactic-scale low-frequency gravitational wave observatories sensitive to the nanohertz frequency band. The primary source of gravitational radiation in this regime is expected to be a stochastic background, formed from the cosmic population of supermassive black hole binaries. Here we present the results obtained by analyzing the 12.5-year data release from the North American Observatory for Gravitational Waves (NANOGrav). Our analysis shows a strong preference for a stochastic process with a common spectrum in all pulsars. However, we find no statistically significant evidence that this process has quadrupolar spatial correlations, which we would consider necessary to claim a gravitational wave background detection. We present a suite of statistical checks to evaluate the significance of these detection metrics, and examine potential implications for the supermassive black hole binary population under the hypothesis that the signal is indeed astrophysical in nature.