link] He considered it a "missing" circuit component that would be non-linear, passive, two-terminal, electrical component. The name is a portmanteau of memory and resistor. The component if ever invented, would not provide constant electrical resistance. It's resistance would be "remembered" and vary depending on the history of current that previously passed through it. This could vary depending on how much current or by direction. It's symbol is above. More here.
Years later Leon backed off that definition and generalized that the memristor was a general and not specific component. He re-imagined his original idea to include all forms of two-terminal non-volatile memory circuit components. In it's new vagueness he now envisioned it as predating the resistor, capacitor and inductor. Whatever. His original idea was at least a highly useful device even if it didn't yet exist.
In the last several years a number of companies have claimed to have at last invented the memristor. These components use anything from titanium dioxide films to CMOS, hafnium oxide, MRAM, RRAM... etc. These are not cranks and dot coms. HP is working on the technology. Critics of the original requirements point out that they are impossible under the known laws of non-equilibrium thermodynamics. Other more cynical critics pointed out that skin and blood meet the criteria. More here. Still no such product is commercially available.
But trolling geeks have pointed out that you can kind of cheat to meet the requirements...