An analysis based on 124000 selected tau pairs recorded by the ALEPH detector at LEP provides the vector (V) and axial-vector (A) spectral functions of hadronic tau decays together with their total widths. This allows the evaluation of finite energy chiral sum rules that are weighted integrals over the (V - A) spectral functions. In addition, a precise measurement of alpha(s) along with a determination of nonperturbative contributions at the tau mass scale is performed. The experimentally and theoretically most robust determination of alpha(s)(M-tau(2)) is obtained from the (V + A) fit that yields alpha(s)(M-tau(2)) = 0.334 +/- 0.022 giving alpha(s)(M-Z(2)) = 0.1202 +/- 0.0027 after the extrapolation to the mass of the Z boson. The approach of the Operator Product Expansion (OPE) is tested experimentally studying the evolution of the tau hadronic widths to masses smaller than the tau mass.