A method is developed for the theoretical investigation of the structure of fluids comprised of hard nonspherical molecules carrying electric charge. The development is based upon the mean spherical approximation for the direct correlation function of such fluids. Since the equations based upon this approximation cannot in general be solved analytically an approximate ansatz for the direct correlation function containing a small number of free parameters is introduced. The free parameters are then determined from a standard variational principle. The ansatz itself is chosen to produce the known results in the strong coupling (large charge) limit and in the special case of hard spheres. In order to ensure that the solution in the limit of strong coupling has a tractable analytic form a new model is proposed for the charge distribution on an individual molecule. The method is applied in detail to a model of charged hard ellipsoids which is a generalization of the primitive model for ionic fluids and is found to be practical; it reduces to the primitive model in the special case of charged hard spheres. It is shown that the approximation preserves the conservation of charge. Properties investigated include the direct correlation function itself, the electrostatic energy of the fluid, the pair distribution function and the electrical potential surrounding an individual molecule. Results for these quantities are obtained for a range of densities and charge covering four orders of magnitude and for molecules with elongations (ratios of lengths of axes) from 0.5 (oblate) to 10 (prolate). The direct correlation function itself is given in an analytic form which can be used together with an appropriate density functional to investigate the structure of the electrical double layer formed by the fluid at a solid boundary.