The predator-prey model with constant harvesting (or stocking) was investigated, and the impact of fear effect on the dynamical behavior of the system was considered. Firstly, the equilibrium points and their local stability were analyzed, and the boundary equilibrium would become a degenerate saddle-node under certain conditions was verified. Secondly, by introducing a series of transformations and utilizing normal form theory, it was proved that the system could undergo saddle-node bifurcation, Hopf bifurcation and degenerate Hopf bifurcation under certain conditions. The theoretical results indicate that for some parameter values the system has exactly two limit cycles, of which the outer limit cycle is stable. Finally, the phase portraits of the system were depicted by MATLAB. Numerical simulations show that a low level of fear can lead to the extinction of populations, that is, an appropriate fear effect is beneficial for the coexistence of two populations.