In this paper, we investigate a nonlinear discrete prey-predator system with fear effects. The existence, local stability and boundedness of positive equilibrium point are discussed. Using the center manifold theorem and bifurcation theory, the conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation in the interior of ${\mathbb{R}}_{+}^2$ are established. Furthermore, the numerical simulations not only show complex dynamical behaviors, but also verify our analysis results. A feedback control strategy is employed to control bifurcation and chaos in the system.