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In this paper we present the results of a systematic investigation of an entire population of starless dust cores within a single molecular cloud. Analysis of extinction data shows the cores to be dense objects characterized by a narrow range of density. Analysis of C18O and NH3 molecular-line observations reveals very narrow lines. The non-thermal velocity dispersions measured in both these tracers are found to be subsonic for the large majority of the cores and show no correlation with core mass (or size). Thermal pressure is thus the dominate source of internal gas pressure and support for most of the core population. The total internal gas pressures of the cores are found to be roughly independent of core mass over the entire range of the core mass function (CMF) indicating that the cores are in pressure equilibrium with an external source of pressure. This external pressure is most likely provided by the weight of the surrounding Pipe cloud within which the cores are embedded. Most of the cores appear to be pressure confined, gravitationally unbound entities whose nature, structure and future evolution are determined by only a few physical factors which include self-gravity, the fundamental processes of thermal physics and the simple requirement of pressure equilibrium with the surrounding environment. The observed core properties likely constitute the initial conditions for star formation in dense gas. The entire core population is found to be characterized by a single critical Bonnor-Ebert mass. This mass coincides with the characteristic mass of the Pipe CMF indicating that most cores formed in the cloud are near critical stability. This suggests that the mass function of cores (and the IMF) has its origin in the physical process of thermal fragmentation in a pressurized medium.