Equilibrium in a two-phase system, a liquid in equilibrium with its vapor, for example, is characterized by the constancy throughout the system of the temperature and the chemical potential and a relation between the pressures of the phases, the Laplace relation of macroscopic thermodynamics. We give a molecular expression for this latter relation in terms of a local pressure defined as the thermodynamic response to a local volume change and we show that this local pressure is the same as the local force distribution in an interface. The macroscopic characteristics of the two-phase system, including the bulk pressures, the surface tension and the location of the surface of tension are all determined by this local pressure function. As shown in a companion paper this function can easily be determined by numerical simulation.