A polarized Raman study of nanographite ribbons on a highly oriented pyrolytic graphite substrate is reported. The Raman peak of the nanographite ribbons exhibits an intensity dependence on the light polarization direction relative to the nanographite ribbon axis. This result is due to the quantum confinement of the electrons in the 1D band structure of the nanographite ribbons, combined with the anisotropy of the light absorption in 2D graphite, in agreement with theoretical predictions.