Memristor theory was formulated and named by Leon Chua in a 1971 paper. Chua strongly believed that a fourth device existed to provide conceptual symmetry with the resistor, inductor, and capacitor. This symmetry follows from the description of basic passive circuit elements as defined by a relation between two of the four fundamental circuit variables. A device linking charge and flux (themselves defined as time integrals of current and voltage), which would be the memristor, was still hypothetical at the time. However, it would not be until thirty-seven years later, on April 30, 2008, that a team at HP Labs led by the scientist R. Stanley Williams would announce the discovery of a switching memristor. Based on a thin film of titanium dioxide, it has been presented as an approximately ideal device.

The reason that the memristor is radically different from the other fundamental circuit elements is that, unlike them, it carries a memory of its past. When you turn off the voltage to the circuit, the memristor still remembers how much was applied before and for how long. That's an effect that can't be duplicated by any circuit combination of resistors, capacitors, and inductors, which is why the memristor qualifies as a fundamental circuit element.

Need For Memristor
A memristor is one of four basic electrical circuit components, joining the resistor, capacitor, and inductor. The memristor, short for “memory resistor” was first theorized by student Leon Chua in the early 1970s. He developed mathematical equations to represent the memristor, which Chua believed would balance the functions of the other three types of circuit elements.
The known three fundamental circuit elements as resistor, capacitor and inductor relates four fundamental circuit variables as electric current, voltage, charge and magnetic flux. In that we were missing one to relate charge to magnetic flux. That is where the need for the fourth fundamental element comes in. This element has been named as memristor.

Memristance (Memory + Resistance) is a property of an Electrical Component that describes the variation in Resistance of a component with the flow of charge. Any two terminal electrical component that exhibits Memristance is known as a Memristor. Memristance is becoming more relevant and necessary as we approach smaller circuits, and at some point when we scale into nano electronics, we would have to take memristance into account in our circuit models to simulate and design electronic circuits properly.
An ideal memristor is a passive two-terminal electronic device that is built to express only the property of memristance (just as a resistor expresses resistance and an inductor expresses inductance). However, in practice it may be difficult to build a 'pure memristor,' since a real device may also have a small amount of some other property, such as capacitance (just as any real inductor also has resistance). A common analogy for a resistor is a pipe that carries water. The water itself is analogous to electrical charge, the pressure at the input of the pipe is similar to voltage, and the rate of flow of the water through the pipe is like electrical current. Just as with an electrical resistor, the flow of water through the pipe is faster if the pipe is shorter and/or it has a larger diameter.