In this thesis, the memory systems of the brain are researched and the relationship between these memory systems and memristor theory are studied. Firstly, the ideal memristor theory is studied. By using the cubic memristor that was based on the ideal memristor theory, the basic logical operations are explained. Additionally, the nerve cells and synapses, which are thought where the memory and learning take place, are inquired. Bearing in mind that the characteristic of the memristor and its similarities between the synapses has been researched with the previous studies on this field. Finally, memristor data storage system has been designed by using memristors, and a binary image of 12x60 pixels has been successfully stored on this design. Also, the edge detection for images has been presented by using the memristor cellular automaton and some examples have also been given.