We present here a detailed theory of electronic surface quantum states in a low magnetic field, as well as of their effects on the microwave surface impedance. A marked oscillatory structure in the microwave absorption as a function of magnetic field has been carefully observed by Khaikin and by Koch et al. The quantized magnetic surface levels are bound states of electrons trapped against the surface by the magnetic field. Even though these levels are somewhat analogous to Landau levels, they have considerably different properties. Resonant transitions between these levels give rise to a series of spectral lines in the surface impedance, just as cyclotron resonance is a result of transitions between Landau levels. The present effect is essentially quantum in nature, however. A considerable amount of quantitative information can be extracted from the experimental data. The Fermi velocity, radius of curvature of the Fermi surface, and mean free time at certain points on the Fermi surface can be obtained. Most novel, however, is the fact that one can extract information on the scattering of electrons by the surface, as a function of impact angle.