There is mounting evidence that active galactic nuclei (AGN) selected through optical emission lines or radio luminosities comprise two distinct AGN populations, whose activity is triggered by different processes. In two previous papers, we studied the host galaxies and environment of radio-loud AGN. In this third paper we study the properties of a sample of Type-2 AGN that were selected on the basis of their [2-10]keV X-ray luminosity. We find that the X-ray luminosity function is in good agreement with previous studies and that the fraction of galaxies hosting an X-ray AGN is a strong function of the stellar mass of the host galaxy. The shape of this fraction-mass relation is similar to the fraction of galaxies that are emission-line AGN, while it differs significantly from the relation observed for radio-selected AGN. The AGN in our sample tend to be located in underdense environments where galaxy mergers and interactions are likely to occur. For all host galaxy masses, the Type-2 AGN display a strong infrared excess at short (~3.5um) wavelengths, suggesting the presence of hot dust possibly associated with a hot dusty torus. These results add weight to the belief that the X-ray selection criteria identifies a population of AGN similar to the emission-line selected population but distinct from the radio population at high masses.