We provide explicit estimates for the small and large eigenvalues of the one-dimensional Schrödinger operator $L(q)$ with a periodic, real and locally integrable potential $q$. For this purpose, we first write explicitly the first and second terms of the asymptotic formulas for large periodic and antiperiodic eigenvalues and update these asymptotic formulas for the Kronig-Penney model. Then, we give estimates for the small periodic and antiperiodic eigenvalues and the lengths of the first gaps in the spectrum of $L(q)$ in the case of the Kronig-Penney model. Moreover, we provide error analysis and present a numerical example.