A two-fluid analysis of the current driven tearing instability is presented. It concentrates on the systematic investigation of the physics related to the plasma compressibility and to the contribution of the Hall term to the inductive electric field, for arbitrary values of the ion skin depth and of the magnitude of the magnetic guide field. The plasma compressibility is treated consistently for a wide range of the plasma beta that excludes only the extremely cold limit where the mode growth rate would become sonic or supersonic. Conversely, the effects associated with the finite ion gyroradius and the equilibrium density and temperature gradients are neglected. Seven parametric regions are identified, characterized by the relative strengths of the Hall and beta parameters. Five of them are amenable to asymptotic analyses yielding analytic dispersion relations and one allows a semi-analytic treatment. The singular, multi-layer structure of the tearing mode and the conditions under which the different components of the magnetic field diffuse are shown in detail for each of those parametric regions.