We analyze very deep X-ray and mid-IR surveys in common areas of the Lockman Hole and the Hubble Deep Field-North (HDF-N) to study the sources of the X-ray background (XRB) and to test the standard obscured accretion paradigm. Observations with XMM-Newton and Infrared Space Observatory (ISO) of a substantial area in Lockman are particularly important to sample luminous-but relatively uncommon-obscured active galactic nuclei (AGNs). We detect a rich population of X-ray luminous sources with red optical colors, including a fraction identified with extremely red objects (R-K>5) and galaxies with spectral energy distributions (SEDs) typical of normal massive ellipticals or spirals at z~1. The X-ray luminosities of these objects (L0.5-10keV~1043-1045 ergs s-1) indicate that the ultimate energy source is gravitational accretion, while the X-ray-to-IR flux ratios and the X-ray spectral hardness show evidence of photoelectric absorption at low X-ray energies. An important hint on the physics comes from the mid-IR data at 6.7 and 15 μm, which are well reproduced by model spectra of completely obscured quasars under standard assumptions and line-of-sight optical depths τ0.3μ~=30-40. Other predictions of the standard X-ray background (XRB) picture, such as the distributions of intrinsic bolometric luminosities and the relative fractions of type I and type II objects (1:3), are also consistent with our results. Obscured gravitational accretion is then confirmed as being responsible for the bulk of the X-ray background, since we detect in the IR the downgraded energy photoelectrically absorbed in X-rays: 63% of the faint 5-10 keV XMM sources are detected in the mid-IR by Fadda et al. As discussed there, however, although as much as 90% of the X-ray energy production could be converted to IR photons, no more than 20% (and possibly less) of the cosmic IR background can be attributed to X-ray-loud AGNs. Based partly on observations obtained with the Infrared Space Observatory and with XMM-Newton ESA science missions with instruments and contributions funded by ESA Member States and the USA (NASA).