The memristor is is just a way to model nonlinear circuit elements and is one of many components in a nonlinear expansion for circuit modeling. See this paper
by Leon Chua, the memristor's inventor. Note that in this paper the fourth element of the four element torus is negative resistance and not the memristor. All of the publicity over the memristor has been (sucessfull) marketing by some researchers at HP.
From the talk page for the memristor on wikipedia
"Resistance, Capacitance and Inductance are regarded as fundamental because to each there corresponds a different picture of what is going on with the energy. Resistance refers to the loss of energy to Joule heating. Capacitance refers to storage of energy in the electric field. Inductance refers to storage of energy in the magnetic field.
If memristance is the "fourth fundamental" circuit element then memristors must do something with the energy they are imparted other than turn it into heat, or store it in electric or magnetic fields. So what do memristor supporters have to say about this? nothing. This is not surprising, since the concept of memristance stems from a purely mathematical argument bent on taming the current/voltage relationships of nonlinear circuit elements. The concept of memristance was invented out of convenience to avoid dealing with frequency-dependent (time-dependent) resistance, inductance, and capacitance. Thus the memeristor is not "fundamental", unless in your book fundamental is synonymous with convenient."