That is the title of the Nature article (Nature 451, 977-980 (21 February 2008)) that I wrote about here. At the time I was complaining about the way this BBC article had described the science involved.
So I had a look at the original Nature article, and agree that it is a wonderful and novel substance, so BBC picked up on a good thing. Let me explain it as I understand it. I hope I'm not too far wrong in this, please correct me if I am.
A normal rubber consists of longs chains of atoms, held together by so-called cross-links. The chains like to be coiled up, but if you pull on them, they can uncoil, so the material stretches. If you let go, it will go back, simply because it prefers to be in its coiled-up state*. Since the chains are cross-linked together, it remembers better than other materials how it was before, so it is more elastic than normal materials. Without the cross-links, it would be a plastic, which does not remember its shape as well.
If you cut a rubber, you break some of the bonds. These are covalent bonds, bonds that have strength because they share electrons. Once a covalent bond is broken, one half of the chain takes more electrons than the other, so this half is negatively charged and the other half is positively charged. Positive and negative charges cancel out at the drop of a hat, and they get rid of their charge as quickly as possible. They might grab a gas molecule from the air or react with another part of the rubber, so in a tiny fraction of a second, the loose ends have been tied up and the cut has sealed over. After it has closed up, it's stable and happy, and will not react again. So when you bring the other bit of the rubber back, it will not go back together, even if you do it as quickly as you can posibly imagine.
This rubber is different, because it is made of short molecules, that are all held together by hydrogen bonds. Hydrogen bonds hold the structure together, but they are not quite as strong as covalent bonds. Just like a normal rubber, the end effect is that a network of bonds is created, where segments can stretch and coil up. So this is still a rubber in terms of how it acts normally. The benefit is that a broken hydrogen bond does not leave positive and negative charges. It will tend to attract molecules to tie up the loose ends, but this is not a quick process. This gives you time to put the rubber back together. Slight movement (diffusion) of the chain ends allow the molecules to line up and bonds back together, so that the material heals and the seal is as good as the original rubber if you give it long enough.
There are polymers (materials that come in long chains of atoms) that can re-heal by diffusion already. These chains do not have cross-links, so they are not so rubbery. To heal themselves, the chains just wriggle together until they are knotted together. That works pretty well, entanglements are the way that loads of polymers hold themselves together anyway. But a material like that is able to flow like honey or jelly (two really good foods)**, so it is not a real rubber.
So the material that they have created is unique and special, in being a useful rubber and being able to heal itself. I'm sure there will be lots of places where it can be used, although the authors do not suggest any themselves (they are more interested in the science, as I am too). The same issue of Nature also contained the comment from Justin Mynar and Takuzo Aida “The potential applications are manifold. Tears in clothes that effectively stitch themselves together, long-lasting coatings and paints for houses and cars, and to take one example on the medical front, self-repairing artificial bones and cartilage.” To me, the possibilities seem endless, and given that it is cheap and easy to make, and easy to degrade (so it can be disposed of environmentally), I think we will see a lot of this in the future.
If this article did not make sense, try Physics World. The Times also enjoyed thinking up uses for it.
* it's all about entropy - coiled up structures have a higher disorder so they are preferred, it's a law of physics
**depending on how liquid it is, people talk about creep or viscosity when they talk about a material flowing (under its own weight or otherwise)