Transverse Anderson localization of light in one-dimensional waveguide arrays with both width and position disorders of individual waveguide has been studied and characterized quantitatively in the viewpoint of transverse localization of eigen modes of the disordered arrays. With the increase in the disorder of the waveguide arrays, more and more eigen modes become localized near the band edges of the finite-sized waveguide arrays in the linear regime. Such a disorder-induced light localization is more effective in waveguide arrays with the width disorder than those with the position disorder. It is shown that, a self-focusing nonlinearity results in an increase of the inverse participation ratio of the nonlinear disordered modes originating from Anderson modes near the edge of the semi-infinite bandgap, therefore, tends to enhance the transverse localization of light. On the other hand, with the increase of the self-defocusing nonlinearity, the inverse participation ratio first increases to a maximum and then decreases for the nonlinear disordered modes originating from Anderson modes near the edge of the first bandgap. The delocalization effect in the self-defocusing case is mainly the result of the joint effects of the normal diffraction near the second band edge and the resonant interaction between the nonlinear disordered modes and the eigen modes in the second band of the waveguide array.