Fractional-order electrical properties are ubiquitous in physical devices, and investigating the features of fractional-order elements (FOEs) is imperative. This paper presents a comprehensive investigation of a generalized FOE model with nonlinear fractional-order constitutive relation. We analyze the response characteristics of FOEs among the four basic integer-order electrical elements, and derive analytic expressions for the step input, zero input, and sinusoidal input responses. The step input response consists of a sequence of power functions, indicating that nonlinear FOEs exhibit an extended power-law step response. The zero input response manifests in three different modes: impulsive, discontinuous, and continuous. Moreover, the sinusoidal input response can be expressed as a series of harmonic additions with calculable amplitude and phase, suggesting a constant phase-shift characteristic of nonlinear FOEs. Our numerical examples demonstrate that FOEs with nonlinear constitutive relations exhibit more intricate dynamic characteristics than linear elements. Additionally, we propose an equivalent analog circuit to achieve the FOE model approximately. The circuit simulation results validate the equivalent circuit's effectiveness and the theoretical conclusion's accuracy. This generalized FOE model extends the existing integer-order nonlinear circuit theory and simplifies the modeling of physical devices in the real world.