The memristor is a new, fourth type of integrated circuits. It gets its name because of how it acts like a variable resistor. In a memristor, there are two strips of metal. When current is run through it, depending on the resistance, the distance between these two strips changes. What is so great about this you ask? Well, when you turn it off, this distance stays the same, and when you run current through it again you can tell how far apart the two strips are.
Currently RAM, (Random Access Memory) only has two states, on or off. Another problem is that these 1’s and 0’s, only keep their state while power is being fed through it. This causes the very long reboot times these days as files from the Hard Drive are re-read and loaded into memory.
With a memristor, there could be near infinate distance between the two bands, thus greatly increasing how much memory can be stored in a given amount of space. Also, the ability to keep that position could allow us to have near-instant re-boots of the system, since the memory will stay even when power is turned off.
The problem however, as you imagine is size. As we progress to nano-electronics, the memristor is still relativley large in comparison. This has kept the memristor from becoming the new commercial memory type. Once this problem is solved however, we will vastly be able to increase the memory and storage space of our drives, and it will lead to a whole new era of both comuting and computer software, no to mention the huge amounts of power we will be able to save.