This paper presents a decentralized adaptive backstepping controller to dampen oscillations and improve the transient stability to parametric uncertainties in multimachine power systems. The proposed design on the $i^{th}$synchronous generator uses only local information and operates without the need for remote signals from the other generators. The design of the nonlinear controller is based on a modified fourth-order nonlinear model of a synchronous generator, and the automatic voltage regulator model is considered so as to decrease the steady state voltage error. The construction of both the control law and the associated Lyapunov function is systematically designed within the design methodology. A 3-machine power system is used to demonstrate the effectiveness of the proposed controller over two other controllers, namely a conventional damping controller (power system stabilizer) and one designed using the feedback linearization techniques.