A kind of predator-prey system with double time delays i.e. the hunting delay (the time delay of the predator maturation) τ₁ and the growth time delay of the predator and the prey τ₂ is investigated. Stability analysis on the equilibrium point of the system is carried out, and the ranges of the time delays are determined. Thus, the impact of the predator and the prey is in-depth studied. First of all, the hunting delay (the time delay of the predator maturation) τ₁ and the growth time delay of the predator and the prey τ₂ are taken as parameters, respectively. The range of the roots with a strictly negative real part is found by analyzing the distribution of characteristic roots for the linearized equation. The condition to ensure the stability of the zero solution of the system and its stable domain are decided. Secondly, when the hunting delay τ₁ is in the stable region, the stability and the existence of Hopf bifurcation are given. Then the Hopf bifurcation is discussed by using the center manifold theory and normal form method introduced by Hassard and Kazarinoff. Both direction and stability of the Hopf bifurcation are considered, and the ranges for the system parameters are obtained.