This paper proposes a novel adaptive nonsingular fast terminal sliding mode control (ANFTSMC) scheme based on a radial basis function neural network (RBFNN) for a quadrotor system in the presence of input saturation, model uncertainties, and external disturbances. A new barrier Lyapunov function and an adaptive law based on RBFNN are utilized for the design of the ANFTSMC, which guarantees that the closed-loop system asymptotically tracks the desired trajectories in a finite time. The model uncertainties are estimated by the adaptive law based on RBFNN, and the chattering phenomenon, a major drawback of conventional sliding mode controllers, is significantly reduced. Additionally, an auxiliary system is proposed to ensure the stability of the quadrotor system under input saturation conditions. The stability of the improved control scheme is proven using Lyapunov stability theory. Finally, numerical simulations in MATLAB are conducted to verify the effectiveness and superior performance of the proposed method compared to existing methods for the trajectory tracking control of a quadrotor system in the presence of input saturation, model uncertainties, and external disturbances.