Memristor is a newly emerged device in nano-technology that provides low power consumption as well as high density. In this paper, the staircase memristor which has multiple stable resistive states is modelled and exploited, and its possible applications are investigated. Comparing with the analog memristor model proposed by HP, the staircase memristor model shows a significant delayed-switching effect which yields a range of more or less discrete memristance values rather than continuous values. We present using the staircase memristor to produce staircase waveforms which can be used in testing circuit or sampling oscilloscope. Besides, staircase memristors are used in cellular neural network (CNN) circuits to represent the connection weights between CNN cells. Benefit from the simple structure, non-volatility and low power properties, the complexity and size of CNN circuits can be reduced.