A molecular-thermodynamic theory is developed to model the micellization of ionic surfactants with added electrolytes in aqueous solution, by combining a thermodynamic description of the micellar solution free energy with a molecular model of the micellization process. The molecular model of micellization, which accounts for the various free-energy contributions associated with assembling the charged micelle from its constituent surfactant ions, allows for a fraction of the counterions, released by the surfactant heads and/or any added electrolytes, to bind onto the micelle surface and effectively induce a partial reduction of the micelle surface charge. The bound counterions are modeled as being intercalated among the surfactant heads on the micelle surface of charge. The remaining counterions are distributed according to the Boltzmann equation in the diffuse region, which lies beyond the Stern layer of steric exclusion adjacent to the micelle surface of charge. Expressions for the various free-energy contributions to micelle formation are derived for the general case when multiple counterion species are present in the solution, including inorganic counterions, such as Na+, as well as organic counterions having pendant hydrophobic groups that penetrate into the micelle core, such as the salicylate ion. In our theoretical formulation, the optimal degree of binding of each counterion species onto the charged micelle surface is a predicted quantity, obtained by minimizing the free energy of micelle formation, and depends on the nature of the counterion, including its hydrated size, valence, and lipophilicity. The theory developed is validated by implementing it in the case of the well-studied ionic surfactant system: sodium dodecyl sulfate (SDS) with added NaCl. We find good quantitative agreement between our theoretical predictions of (i) the optimal degree of Na+ binding onto the charged micelle surface, (ii) the electrostatic potential at the micelle surface of charge, (iii) the electrostatic potential at the Stern surface, and (iv) the critical micelle concentration (cmc) and the available experimental results for these properties. In addition, the optimal degree of counterion binding is predicted to vary with the bulk electrolyte concentration and with the micelle shape, consistent with the experimentally observed variations of the counterion concentration in the micelle-water interfacial region. In paper 2 of this series, the molecular -thermodynamic theory developed here is utilized to predict various micellar solution properties, including the cmc, the optimal degree of counterion binding, and the average micelle aggregation numbers, of aqueous solutions of ionic surfactants containing monovalent, multivalent, and/or organic counterions.