The Carrington event in 1859, a solar flare with an associated geomagnetic storm, has served as a prototype of possible superflare occurrence on the Sun. Recent geophysical (14C signatures in tree rings) and precise time-series photometry [the bolometric total solar irradiance (TSI) for the Sun, and the broadband photometry from Kepler and Transiting Exoplanet Survey Satellite, for the stars] have broadened our perspective on extreme events and the threats that they pose for Earth and for Earth-like exoplanets. This review assesses the mutual solar and/or stellar lessons learned and the status of our theoretical understanding of the new data, both stellar and solar, as they relate to the physics of the Carrington event. The discussion includes the event's implied coronal mass ejection, its potential "solar cosmic ray" production, and the observed geomagnetic disturbances based on the multimessenger information already available in that era. Taking the Carrington event as an exemplar of the most extreme solar event, and in the context of our rich modern knowledge of solar flare and/or coronal mass ejection events, we discuss the aspects of these processes that might be relevant to activity on solar-type stars, and in particular their superflares. The Carrington flare of 1859, though powerful, did not significantly exceed the magnitudes of the greatest events observed in the modern era. Stellar "superflare" events on solar-type stars may share common paradigms, and also suggest the possibility of a more extreme solar event at some time in the future. We benefit from comparing the better-known microphysics of solar flares and coronal mass ejections with the diversity of related stellar phenomena.