Continuous-time memristors have been used in numerous chaotic circuit systems. Similarly, the discrete memristor model applied to a discrete map is also worthy of further study. To this end, this paper first proposes a discrete memristor model and analyzes the voltage-current characteristics of the memristor. Also, the discrete memristor is coupled with a one-dimensional (1D) sine chaotic map through different coupling frameworks, and two different two-dimensional (2D) chaotic map models are generated. Due to the presence of linear fixed points, the stability of the 2D memristor-coupled chaotic map depends on the choice of control parameters and initial states. The dynamic behavior of the chaotic map under different coupled map frameworks is investigated by using various analytical methods, and the results show that different coupling frameworks can produce different complex dynamical behaviors for memristor chaotic maps. The dynamic behavior based on parameter control is also investigated. The numerical experimental results show that the change of parameters can not only enrich the dynamic behavior of a chaotic map, but also increase the complexity of the memristor-coupled sine map. In addition, a simple encryption algorithm is designed based on the memristor chaotic map under the new coupling framework, and the performance analysis shows that the algorithm has a strong ability of image encryption. Finally, the numerical results are verified by hardware experiments.