A measurement of the W boson pair production cross section in proton-proton collisions at √s = 8 TeV is presented. The data collected with the CMS detector at the LHC correspond to an integrated luminosity of 19.4fb-1. The W +W - candidates are selected from events with two charged leptons, electrons or muons, and large missing transverse energy. The measured W +W - cross section is 60.1 ± 0.9 (stat) ± 3.2 (exp) ± 3.1 (theo) ± 1.6 (lumi) pb = 60.1±4.8 pb, consistent with the standard model prediction. The W +W - cross sections are also measured in two different fiducial phase space regions. The normalized differential cross section is measured as a function of kinematic variables of the final-state charged leptons and compared with several perturbative QCD predictions. Limits on anomalous gauge couplings associated with dimension-six operators are also given in the framework of an effective field theory. The corresponding 95 % confidence level intervals are -5.7<cWWW/Λ2<5.9TeV-2, -11.4<cW/Λ2<5.4TeV-2, -29.2<cB/Λ2<23.9TeV-2, in the HISZ basis.