We present NuSTAR observations of the powerful radio galaxy Cygnus A, focusing on the central absorbed active galactic nucleus (AGN). Cygnus A is embedded in a cool-core galaxy cluster, and hence we also examine archival XMM-Newton data to facilitate the decomposition of the spectrum into the AGN and intracluster medium components. NuSTAR gives a source-dominated spectrum of the AGN out to $\gt 70$ keV. In gross terms, the NuSTAR spectrum of the AGN has the form of a power law (${\rm{\Gamma }}\sim 1.6-1.7$) absorbed by a neutral column density of ${N}_{{\rm{H}}}\sim 1.6\times {10}^{23}\;\;{\mathrm{cm}}^{-2}$. However, we also detect curvature in the hard ($\gt 10$ keV) spectrum resulting from reflection by Compton-thick matter out of our line of sight to the X-ray source. Compton reflection, possibly from the outer accretion disk or obscuring torus, is required even permitting a high-energy cut off in the continuum source; the limit on the cut-off energy is ${E}_{\mathrm{cut}}\gt 111$ keV(90% confidence). Interestingly, the absorbed power law plus reflection model leaves residuals suggesting the absorption/emission from a fast ($15,000-26,000\;\;\mathrm{km}\;\;{{\rm{s}}}^{-1}\;$), high column-density (${N}_{W}\gt 3\times {10}^{23}\;\;{\mathrm{cm}}^{-2}$), highly ionized ($\xi \sim 2500\;\mathrm{erg}\;\mathrm{cm}\;{{\rm{s}}}^{-1}$) wind. A second, even faster ionized wind component is also suggested by these data. We show that the ionized wind likely carries a significant mass and momentum flux, and may carry sufficient kinetic energy to exercise feedback on the host galaxy. Show Partial Abstract