Unlike other branches of astronomy that have proprietary telescopes, X-ray astronomy is of necessity done using satellites and suborbital rockets. These are funded by national governments, which typically insist that all data be made public within a reasonable time. As a result, practically all X-ray-astronomy observations are available on-line; the challenge is to find the right data. Fortunately, X-ray-astronomy data are concentrated in a small number of archives and it is usually clear which website to try.
The best way to think of an archive is as a collection of tables, some of which have data sets attached. Some of the tables are simple catalogs, e.g. a list of stars with positions, spectral types, and fluxes. Other tables come with considerable data attached, e.g. the observation catalog for some mission that lists pointing position, exposure time and so forth but also provides links to all the publically available data for the observation. These data may include basic event files, cleaned event files, auxiliary information such as housekeeping and orbit files, and product files such as spectra, images, and lightcurves. The archive Internet interface will generally allow the astronomer to choose which categories of data to download for the selected observation.
As an example, consider finding Chandra observations of the Perseus cluster. Its name can be entered in the Chandra data archive search page and the NED or SIMBAD servers used to translate the name into a position.