(Abridged) We report results from a comprehensive study of the nearby M dwarf Proxima Centauri with the XMM-Newton satellite. We find strongly variable coronal X-ray emission, with flares ranging over a factor of 100 in peak flux. The low-level emission is found to be continuously variable. Several weak flares are characteristically preceded by an optical burst, compatible with predictions from standard solar flare models. We propose that the U band bursts are proxies for the elusive stellar non-thermal hard X-ray bursts suggested from solar observations. A very large X-ray flare was observed in its entirety, with a peak luminosity of 3.9E28 erg/s [0.15-10 keV] and a total X-ray energy of 1.5E32 erg. This flare has allowed to measure significant density variations from X-ray spectroscopy of the OVII He-like triplet: we find peak densities reaching up to 4E11 cm^-3 for plasma of about 1-5 MK. Abundance ratios show little variability in time, with a tendency of elements with a high first ionization potential to be overabundant relative to solar photospheric values. We do not find significant effects due to opacity during the flare, indicating that large opacity increases are not the rule even in extreme flares. We model the large flare in terms of an analytic 2-Ribbon flare model and find that the flaring loop system should have large characteristic sizes (~1R*). These results are supported by full hydrodynamic simulations. Comparing the large flare to flares of similar size occurring much more frequently on more active stars, we propose that the X-ray properties of active stars are a consequence of superimposed flares such as the example analyzed in this paper. Such a model also explains why, during episodes of low-level emission, more active stars show hotter plasma than less active stars.