In this paper, an adaptive tracking controller based on a three-layer neural network (NN) with an online weight tuning algorithm is proposed for a nonholonomic wheeled mobile robot in the presence of unknown wheel slips, model uncertainties, and unknown bounded disturbances. The online weight tuning algorithm is modified from the backpropagation with an e-modification term required to assure that the NN weights are bounded. Preliminary neural network offline training is not essential for the weights. Thanks to this proposed controller, the desired tracking performance is achieved where position tracking errors converge to an arbitrarily small neighborhood of the origin regardless of their initial values. According to Lyapunov theory and LaSalle extension, the stability of the whole closed- loop system is ensured to obtain the desired tracking performance. Computer simulations are implemented to certify the validity of the proposed controller.