Model systems of charged spherical macroions and point counterions interacting solely through hard-sphere and Coulomb interaction were investigated by means of extensive Monte Carlo simulations. The macroion-charge to counterion-charge ratio was varied from 10 to 80, the macroion volume fraction from 0.001 25 to 0.08, and the reduced parameter that remained, an electrostatic coupling parameter, over a range of two orders in magnitude. The part of the parameter space investigated includes nearly all experimental colloidal solutions (in the salt-free limit) in which the colloids are carrying up 80 (monovalent counterions) or 160 (divalent counterions) elementary charges. The effects on the structure of varying the counterion charge, the macroion charge, the macroion size, the temperature, the dielectric permittivity, and the macroion volume fraction are presented. At a low charge ratio, a low volume fraction, and/or a low electrostatic coupling, the counterion distribution is only weakly perturbed by the macroions, the system being nearly ideal. At a higher electrostatic coupling, the counterions accumulate near the macroions and screen the electrical field of the macroions. At a very high electrostatic coupling, the system displays a thermodynamic instability and separates into two phases of widely differing electrolyte concentration. Parts of the binodal surface were determined and critical points were estimated. The concept of effective macroion charge is discussed.