We present the first version of the Automatic Learning for the Rapid Classification of Events (ALeRCE) broker light curve classifier. ALeRCE is currently processing the Zwicky Transient Facility (ZTF) alert stream, in preparation for the Vera C. Rubin Observatory. The ALeRCE light curve classifier uses variability features computed from the ZTF alert stream and colors obtained from AllWISE and ZTF photometry. We apply a balanced random forest algorithm with a two-level scheme where the top level classifies each source as periodic, stochastic, or transient, and the bottom level further resolves each of these hierarchical classes among 15 total classes. This classifier corresponds to the first attempt to classify multiple classes of stochastic variables (including core- and host-dominated active galactic nuclei, blazars, young stellar objects, and cataclysmic variables) in addition to different classes of periodic and transient sources, using real data. We created a labeled set using various public catalogs (such as the Catalina Surveys and Gaia DR2 variable stars catalogs, and the Million Quasars catalog), and we classify all objects with 6 g-band or 6 r-band detections in ZTF (868,371 sources as of 2020 June 9), providing updated classifications for sources with new alerts every day. For the top level we obtain macro-averaged precision and recall scores of 0.96 and 0.99, respectively, and for the bottom level we obtain macro-averaged precision and recall scores of 0.57 and 0.76, respectively. Updated classifications from the light curve classifier can be found at the ALeRCE Explorer website (http://alerce.online).