An atom absorbs an X-ray when the photon energy is sufficient to eject a photoelectron (see the XPS document for an energy level diagram). Below this threshold energy there is no absorption. Photons with energies greater than the threshold energy to produce a photoelectron are absorbed because the excess energy is conserved by transferring it to kinetic energy of the photoelectron. However, the probability of the absorption occuring decreases as the photon energy increases above the threshold.
Use of X-ray absorption as an analytical method is fairly uncommon because other techniques such as X-ray fluorescence are more sensitive. The absorption of X-rays by a certain element is often used in analytical instrumentation as a filter to block some X-ray wavelengths. For example, the absorption edge of Zr will block KB and most of the continuum radiation of X-rays from a Mo target. Extended X-ray Absorption Fine Structure (EXAFS)
Extended X-ray Absorption Fine Structure (EXAFS) occurs due to interference effects as photoelectrons leave the surface of a material. The interference depends on the interatomic distance between the atom that ejected the photoelectron and the nearest neighbor atom, and to lesser extents the next nearest neighbor and other atoms. EXAFS therefore provides a means to determine the structure of the atoms on the surface of a material. The fine structure appears as a modulation on the absorption edge, and requires sophisticated modeling to extract the structural information about the sample surface. The most common source for EXAFS is synchrotron radiation, which provides an intense and tunable source of X-rays.