The focus of the present paper is on the detailed description of the internal structures of low mass, population II stars, to clarify some issues about these stellar models and, mainly, their present reliability for observational comparisons. We then explore 1) the role of the local convective model; 2) the differences between "grey" and "non grey" models, and between models in which the photospheric boundary conditions are set at different optical depths (τph = 3 or 100); 3) the role of the equation of state (EoS), both in the atmospheric models and in the interior. One of the major conclusions of the paper is a cautionary note about the usage of the additive volume law in EoS calculations. The dependence of the HR diagram locations and mass luminosity relations on metal and helium content are also discussed. A few comparisons with globular cluster stars show that: 1) general consistency of distance scales and morphologies in the HR diagram is found, when comparing ground based measurements in the Johnson B and V bands and observations in the HST bands; 2) a discrepancy between models and observations may exist for more metal rich clusters; 3) the plausible hypothesis that the mass function in the globular cluster NGC 6397 behaves smoothly until the lower limit of the main sequence poses constraints on the mass-luminosity relation at the lowest end of the main sequence. The evolutionary tracks are available at the WEB location http://www.mporzio.astro.it.