It states that no normal modes can be both Infrared and Raman active in a molecule that possesses a center of symmetry.
This is a powerful application of group theory to vibrational spectroscopy, and allows one to easily detect the presence of this symmetry element by comparison of the IR and Raman spectra generated by the same molecule.
[1] The rule arises because in a centrosymmetric point group, IR active modes, which must transform according to the same irreducible representation generated by one of the components of the dipole moment vector (x, y or z), must be of ungerade (u) symmetry, i.e. their character under inversion is -1, while Raman active modes, which transform according to the symmetry of the polarizability tensor (product of two coordinates), must be of gerade (g) symmetry since their character under inversion is +1.
Thus, in the character table there is no irreducible representation that spans both IR and Raman active modes, and so there is no overlap between the two spectra.
Such spectroscopically "silent" or "inactive" modes exist in molecules such as ethylene (C2H4), benzene (C6H6) and the tetrachloroplatinate ion (PtCl42−).