We report Chandra Advanced CCD Imaging Spectrometer and quasi-simultaneous Rossi X-Ray T iming Explorer (RXT E) observations of the nearby, powerful radio galaxy Cygnus A, with the present paper focusing on the properties of the active nucleus. In the Chandra observation, the hard ([a few keV) X-ray emission is spatially unresolved with a size (1.5 kpc, km s~1 Mpc~1) and coin-[1A H 0 \ 50 cides with the radio and near-infrared nuclei. In contrast, the soft (\2 keV) emission exhibits a bipolar nebulosity that aligns with the optical bipolar continuum and emission-line structures and approximately with the radio jet. In particular, the soft X-ray emission corresponds very well with the [O III] j5007 and Ha ] [N II] jj6548, 6583 nebulosity imaged with Hubble Space T elescope. At the location of the nucleus, there is only weak soft X-ray emission, an e †ect that may be intrinsic or result from a dust lane that crosses the nucleus perpendicular to the source axis. The spectra of the various X-ray components have been obtained by simultaneous Ðts to the six detectors. The compact nucleus is detected to 100 keV and is well described by a heavily absorbed power-law spectrum with (similar to other ! h \ 1.52~0 .12 0.12 narrow-line radio galaxies) and equivalent hydrogen column cm~2. This N H (nuc) \ 2.0~0 .2 0.1 ] 1023 column is compatible with the dust obscuration to the near-infrared source for a normal gas-to-dust ratio. The soft (\2 keV) emission from the nucleus may be described by a power-law spectrum with the same index (i.e., although direct Ðts suggest a slightly larger value for Narrow emission lines ! l \ ! h ), ! l . from highly ionized neon and silicon, as well as a "" neutral ÏÏ Fe Ka line, are detected in the nucleus and its vicinity kpc). The equivalent width (EW) of the Fe Ka line eV) is in good agreement (r [ 2 (182~5 4 40 with theoretical predictions for the EW versus relationship in various geometries. An Fe K edge N H (nuc) is also seen. The RXT E observations indicate a temperature of keV for the cluster gas kT \ 6.9~1 .0 0.3 (discussed in Paper III of this series) and cluster emission lines of Fe Ka and Fe Kb and/or Ni Ka.We consider the possibility that the extended soft X-ray emission is electron-scattered nuclear radiation. Given that 1% of the unabsorbed 2È10 keV nuclear radiation would have to be scattered, the necessary gas column cm~2] would absorb the X-rays rather than scatter [N H (scattering)^3.5 ] 1022 them if the gas is cold. Thus, the scattering plasma must be highly ionized. If this ionization is achieved through photoionization by the nucleus, the ionization parameter m [ 1 ergs cm s~1 and the electron density cm~3 given the observed distance of the soft X-ray emission from the nucleus. The elecn e^6 tron column density inferred from the X-ray observations is much too low to account for the extended optical scattered light, strongly suggesting that the polarized optical light is scattered by dust. The presence of highly ionized Ne lines in the soft X-ray spectrum requires 20...