We derive an analytical theory of the prestellar core initial mass function (IMF) based on an extension of the Press-Schechter statistical formalism. Our approach relies on the general concept of the gravothermal and gravoturbulent collapse of a molecular cloud, with a selection criterion based on the thermal or turbulent Jeans mass, which yields the derivation of the mass spectrum of self-gravitating objects in a quiescent or a turbulent environment. The same formalism also yields the mass spectrum of non-self-gravitating clumps produced in supersonic flows. The mass spectrum of the self-gravitating cores reproduces well the observed IMF. The theory predicts that the shape of the IMF results from two competing contributions, namely, a power law at large scales and an exponential cutoff (lognormal form) centered around the characteristic mass for gravitational collapse. The cutoff exists both in the case of thermal or turbulent collapse, provided that the underlying density field has a lognormal distribution. Whereas pure thermal collapse produces a power-law tail steeper than the Salpeter value, dN/dlogM~M^-x with x~=1.35, the latter is recovered exactly for the (three-dimensional) value of the spectral index of the velocity power spectrum, n~=3.8, found in observations and in numerical simulations of isothermal supersonic turbulence. Indeed, the theory predicts that x=(n+1)/(2n-4) for self-gravitating structures and x=2-n^'/3 for non-self-gravitating structures, where n^' is the power spectrum index of logrho. We show that, whereas supersonic turbulence promotes the formation of both massive stars and brown dwarfs, it has an overall negative impact on star formation, decreasing the star formation efficiency. This theory provides a novel theoretical foundation to understand the origin of the IMF and provides useful guidance to numerical simulations exploring star formation, while making testable predictions.