Two University of Manchester scientists got the Nobel Prize in 2010 for their work with this substance.
The way I understand it is that it is basically a one-atom thick sheet of tightly structured carbon atoms. It is basically a miracle material - 200 times stronger than steel. So, thin that you would have to stack 3 million sheets of it before you got one millimeter of material thickness. Conducts electricity and heat better than any other material - like a superconductor. Some uses:
- Filtering and desalinization - For some reason, the only compound that can pass through it is water. As a result, it has the ability to create drinking water out of sea water at a fraction of the cost of current reverse-osmosis technology. Just last month, Lockheed Martin announced a new a filter based on it. This could be a game-changer for the developing world and countries with problems with potable water.
- Monitors and touch-panel screens - Graphene could allow you to make a screen as thin as wallpaper, and which you could roll up like a piece of paper and take with you.
- Skins and panels for airplanes, cars, etc. - the sheet it creates is incredibly strong. One example was that you could make a hammock out of it that would support a 4 kg cat. But the amount of graphene used would weigh less than one of the cat's whiskers. if you make a plane or car's body out of this material, not only will it be weigh stronger, but it will be a fraction of the weight. Basically lighter, more fuel efficient and safer all at the same time.
- Batteries and transistors - batteries with more of a charge. Smaller transistors for faster computing.
- Solar cells -- super conductivity could make solar cells much more efficient at converting and storing energy
- Graphene is a sustainable and green product because it is made of a plentiful material --- carbon.
The biggest practical issue in bringing graphene products to market has been isolating it as a pure 2D (one-atom-thick) product. Apparently, nature abhors 2D almost as much as it abhors a vacuum, and as a result atoms tend to stack in three dimensions. But it looks like they are getting around that impediment pretty quickly. Here is a page where new graphene products are being announced (Graphene - News & Rumors | ExtremeTech).
And here is a video explaining some advantages of it: