BL Lac, the eponymous blazar, flared to historically high levels at millimeter, infrared, X-ray, and gamma-ray wavelengths in 2012. We present observations made with Herschel, Swift, NuSTAR, Fermi, the Submillimeter Array, CARMA, and the VLBA in 2012-2013, including three months with nearly daily sampling at several wavebands. We have also conducted an intensive campaign of 30 hr with every-orbit observations by Swift and NuSTAR, accompanied by Herschel, and Fermi observations. The source was highly variable at all bands. Time lags, correlations between bands, and the changing shapes of the spectral energy distributions can be explained by synchrotron radiation and inverse Compton emission from nonthermal seed photons originating from within the jet. The passage of four new superluminal very long baseline interferometry knots through the core and two stationary knots about 4 pc downstream accompanied the high flaring in 2012-2013. The seed photons for inverse Compton scattering may arise from the stationary knots and from a Mach disk near the core where relatively slow-moving plasma generates intense nonthermal radiation. The 95 spectral energy distributions obtained on consecutive days form the most densely sampled, broad wavelength coverage for any blazar. The observed spectral energy distributions and multi-waveband light curves are similar to simulated spectral energy distributions and light curves generated with a model in which turbulent plasma crosses a conical shock with a Mach disk.