Lipid bilayers exhibit a phase behavior that involves two distinct, but coupled, order-disorder processes, one in terms of lipid-chain crystalline packing (translational degrees of freedom) and the other in terms of lipid-chain conformational ordering (internal degrees of freedom). Experiments and previous approximate theories have suggested that cholesterol incorporated into lipid bilayers has different microscopic effects on lipid-chain packing and conformations and that cholesterol thereby leads to decoupling of the two ordering processes, manifested by a special equilibrium phase, "liquid-ordered phase," where bilayers are liquid (with translational disorder) but lipid chains are conformationally ordered. We present in this paper a microscopic model that describes this decoupling phenomena and which yields a phase diagram consistent with experimental observations. The model is an off-lattice model based on a two-dimensional random triangulation algorithm and represents lipid and cholesterol molecules by hard-core particles with internal (spin-type) degrees of freedom that have nearest-neighbor interactions. The phase equilibria described by the model, specifically in terms of phase diagrams and structure factors characterizing different phases, are calculated by using several Monte Carlo simulation techniques, including histogram and thermodynamic reweighting techniques, finite-size scaling as well as non-Boltzmann sampling techniques (in order to overcome severe hysteresis effects associated with strongly first-order phase transitions). The results provide a consistent interpretation of the various phases of phospholipid-cholesterol binary mixtures based on the microscopic dual action of cholesterol on the lipid-chain degrees of freedom. In particular, a distinct small-scale structure of the liquid-ordered phase has been identified and characterized. The generic nature of the model proposed holds a promise for a unifying description for a whole series of different lipid-sterol mixtures.