HP invented the memristors (some would even say "discovered") several years ago and managed to create memristors on a dye after that so that memristors chips could be produced.
Now they have announced the release in 2015 of an Operating Systems that will allow developers to create applications based on memristors, making programming them easier.
Memristors are a sort of transistors that can remember what took place (hence the name memory-transistor shrunk into a single word). In a way they are similar to neurones, whose activity depend both on the information (stimuli) received at a given instant and on what happened before. Because of this neutrons can perform interesting computation that are very complex to be implemented using transistors. Not so if one were to use memristors for computation.
Of course as neurones are not good for all kinds of computations, so memristors are not good for all kinds of computations, in a word, they are not going to replace transistors.
If you are looking for a precise result, like how much is it 2+2, you'd expect your computation device to come up with 4 every single time. On the other hand if you are asking a brain for the 4th time in a row how much is 2+2 you are likely to get something like "Are you bloody deaf?"
However, at least for the time being, HP goal is not to create a computer that mimics a brain, rather to have a computer that leverages the memristors characteristics to better process huge amounts of data. They have launched a program called "The Machine".
This, claims HP, will be a radically new computation system, using electrons for processing, photons for communications and ions for storage. Memristors are instrumental in fusing memory with storage and in bringing processing closed to data thus achieving faster data processing. You can get an overview in the video clip.
The Operating Systems 2015 release announcement is a step in this direction.