In this paper, we consider the nonlinear dynamics of the fluid model of transmission control protocol (TCP) network, including the local stability and the Hopf bifurcation. We specify the time delay stabilization range of the system at equilibrium and propose an effective method to determine the direction and stability of the bifurcation producing periodic solutions. In the linear analysis process, the τ-decomposition technique is utilized to deal with the local stability problem of the time delay system, and an explicit expression for the time delay stability region is provided. For the nonlinear analysis part, the formulas for determining the direction of Hopf bifurcation and the stability of the periodic solution are derived with the help of the central manifold theorem and the canonical type theory as the basis, which enable the bifurcation parameters to be calculated directly. Finally, the accuracy and practicality of the theoretical analysis in this paper are verified through a series of typical numerical simulation cases.