We report Chandra Advanced CCD Imaging Spectrometer and quasi-simultaneous Rossi X-Ray Timing 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 less than or similar to1" (1.5 kpc, H-o = km s(-1) Mpc(-1)) and coincides 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] λ5007 and Hα + [N II] λλ6548, 6583 nebulosity imaged with Hubble Space Telescope. At the location of the nucleus, there is only weak soft X-ray emission, an effect 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 fits to the six detectors. The compact nucleus is detected to 100 keV and is well described by a heavily absorbed power-law spectrum with Γ(h) = 1.52(-0.12)(+0.12) (similar to other narrow-line radio galaxies) and equivalent hydrogen column N-H(nuc) = 2.0(-0.2)(+0.1) x 10(23) cm(-2). This 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., Γ(l) = Γ(h)), although direct fits suggest a slightly larger value for Γ(l). Narrow emission lines from highly ionized neon and silicon, as well as a "neutral" Fe Kα line, are detected in the nucleus and its vicinity (r ≲ 2 kpc). The equivalent width (EW) of the Fe Kα line (182(-54)(+40) eV) is in good agreement with theoretical predictions for the EW versus relationship in various geometries. An Fe K edge 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 Kalpha and Fe Kbeta and/or Ni Kalpha. We consider the possibility that the extended soft X-ray emission is electron-scattered nuclear radiation. Given that 1% of the unabsorbed 2-10 keV nuclear radiation would have to be scattered, the necessary gas column [N-H(scattering) similar or equal to 3.5 x 10(22) cm(-2)] would absorb the X-rays rather than scatter 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 xi>1 ergs cm s(-1) and the electron density n(e) similar or equal to 6 cm(-3) given the observed distance of the soft X-ray emission from the nucleus. The electron 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 ergs cm ergs cm s(-1) less than or similar to xi less than or similar to 300 ergs cm s(-1); these lines may originate closer to the nucleus than the extended soft continuum or in a lower density gas. A collisionally ionized thermal model of the extended soft X-rays cannot be ruled out but is unattractive in view of the low metal abundance required (Z = 0.03 Z(.)). The hard X-ray to far-infrared ratio for the nucleus of Cygnus A is similar to that seen in Seyfert 1 and unobscured radio galaxies. By means of the correlation between hard X-ray luminosity and nuclear optical absolute magnitude for these classes of object, we estimate M-B = -22.4 for Cygnus A, near the borderline between Seyfert galaxies and QSOs.