In this paper, a fractional-order locally active memristor is proposed based on the definition of fractional derivative. It is found that the side lobe area of the pinched hysteretic curve of the memristor changes with the fractional-order value, and the side lobe’s area of the fractional-order memristor is greater than that of them emristor with integer order, meaning that the memory of the fractional-order memristor is stronger than that of the memristor with integer order. It is proved by the dynamic rout map (DRM) that the fractional-order memristor possesses continuous memory. The pinched hysteresis, memristance and power characteristics which vary with the fractional order are compared and analysed in detail. Furthermore, we use the memristor to construct a fractional-order chaotic circuit, which can exhibit continuous chaotic motion in the range of 0.75 < fractional order α < 1 and various coexisting attractors. We also show that the lower fractional order causes higher complexity of the fractional order chaotic system using different methods, such as Lyapunov exponent spectrum, bifurcation diagram, spectral entropy and C0 complexity method. Finally, the circuit simulations of the fractional-order chaotic circuit are realised, demonstrating the validity of the mathematical model and the theoretical analysis.