This work focuses primarily on the relative equilibrium and stability of a three-body tethered satellite which center of mass moves along an elliptical orbit with arbitrary eccentricity in the central Newtonian field. We only consider the case with straight tethers, and the lengths of them are small compared to the size of the orbit. It means that the attitude motion of the satellite has no impact on its orbital motion. Additionally, it assumes that the lengths of the tethers are controllable. The law of the tether length control is determined on the basis of the dynamical model of the system so that there exists an Earth-pointing relative equilibrium of the tethered satellite. By analyzing the Floquet multipliers of the linearized system, Lyapunov instability regions and linear stability regions of the relative equilibrium are obtained in a plane of dimensionless parameters. Further nonlinear stability analysis is presented in the linear stability regions by means of a normal form technique and stability theorems on the Hamiltonian system. In particular, stability of nonresonant cases as well as fourth-order resonant cases are investigated in detail.