Characterizing purely elastic or purely viscous rheological nonlinearities is straightforward. However, a comprehensive framework does not exist to characterize materials which exhibit both modes of nonlinearity simultaneously. We develop a descriptive language and set of unambiguous material measures for quantifying the nonlinear viscoelastic response of materials subjected to large amplitude oscillatory shear (LAOS) deformation, enabling us to develop a unique "rheological fingerprint" of an a priori unknown substance. We examine two representative soft materials with our framework; the new material measures clearly reveal properties such as local strain-stiffening which are obscured by conventional means. Our technique is universally applicable to any complex material and provides a test bed for new and existing constitutive models.