A key parameter to the description of all star formation processes is the density structure of the gas. In this letter, we make use of probability distribution functions (PDFs) of Herschel column density maps of Orion B, Aquila, and Polaris, obtained with the Herschel Gould Belt survey (HGBS). We aim to understand which physical processes influence the PDF shape, and with which signatures. The PDFs of Orion B (Aquila) show a lognormal distribution for low column densities until Av 3 (6), and a power-law tail for high column densities, consistent with a rho r^-2 profile for the equivalent spherical density distribution. The PDF of Orion B is broadened by external compression due to the nearby OB stellar aggregates. The PDF of a quiescent subregion of the non-star-forming Polaris cloud is nearly lognormal, indicating that supersonic turbulence governs the density distribution. But we also observe a deviation from the lognormal shape at Av>1 for a subregion in Polaris that includes a prominent filament. We conclude that (i) the point where the PDF deviates from the lognormal form does not trace a universal Av-threshold for star formation, (ii) statistical density fluctuations, intermittency and magnetic fields can cause excess from the lognormal PDF at an early cloud formation stage, (iii) core formation and/or global collapse of filaments and a non-isothermal gas distribution lead to a power-law tail, and (iv) external compression broadens the column density PDF, consistent with numerical simulations.Read less <