We present the first extensive radio to γ-ray observations of a fast-rising blue optical transient, AT 2018cow, over its first ∼100 days. AT 2018cow rose over a few days to a peak luminosity Lpk ∼ 4 × 1044 erg s−1, exceeding that of superluminous supernovae (SNe), before declining as L ∝ t−2. Initial spectra at δt 15 days were mostly featureless and indicated large expansion velocities v ∼ 0.1c and temperatures reaching T ∼ 3 × 104 K. Later spectra revealed a persistent optically thick photosphere and the emergence of H and He emission features with v ∼ 4000 km s−1 with no evidence for ejecta cooling. Our broadband monitoring revealed a hard X-ray spectral component at E ≥ 10 keV, in addition to luminous and highly variable soft X-rays, with properties unprecedented among astronomical transients. An abrupt change in the X-ray decay rate and variability appears to accompany the change in optical spectral properties. AT 2018cow showed bright radio emission consistent with the interaction of a blast wave with vsh ∼ 0.1c with a dense environment ( for vw = 1000 km s−1). While these properties exclude 56Ni-powered transients, our multiwavelength analysis instead indicates that AT 2018cow harbored a "central engine," either a compact object (magnetar or black hole) or an embedded internal shock produced by interaction with a compact, dense circumstellar medium. The engine released ∼1050-1051.5 erg over ∼103-105 s and resides within low-mass fast-moving material with equatorial-polar density asymmetry (Mej,fast 0.3 M☉). Successful SNe from low-mass H-rich stars (like electron-capture SNe) or failed explosions from blue supergiants satisfy these constraints. Intermediate-mass black holes are disfavored by the large environmental density probed by the radio observations.