In this paper, a new time second-order characteristic finite element method is proposed and analyzed for solving the advection-diffusion equations involving a nonlinear right side term. In order to obtain a time second-order characteristic scheme, the global derivative term transferred from the time derivative and advection terms is discretized by using difference operator along the characteristics, the diffusion term is discretized by the average operator along the characteristics, while specially a second-order extrapolation along the characteristics is applied to the right side nonlinear function. We analyze and prove that the proposed scheme for the nonlinear advection-diffusion equations has second order accuracy in time step size, which improves the first order accuracy in time of the classical characteristic methods. The proposed characteristic FEM scheme allows to use large time step sizes in computation. Numerical tests are taken to show the accuracy of our proposed scheme, and the case of single-species population dynamics is further simulated and analyzed by using our method and numerical results show its advantage and effectiveness.