Twistor theory is a mathematical construction that dates back to the sixties, and is probably mostly known for some of its uses within string theory. Funnily enough, it has now been brought to the fore by Peter Woit, a mathematical physicist from Columbia University who became internationally renowned when he published his 2006 book "Not Even Wrong".
In his book Woit exposed the nudity of the then self-proclaimed king of theoretical investigations in fundamental physics, string theory: he showed how all the claims that string theory was the best attempt at yielding a theory of everything were unsubstantiated and basically only hype, as the theory was most likely rather an intriguing piece of mathematics than anything connected with physical reality: 30 years of deep studies and large investments in human resources and excellent minds had not materialized any possible practical, concrete candidate of a real theory of physical reality, or yielded any experimentally testable prediction. Maybe sad, but true, Woit's assessment has stood the test of time.
But Woit is also very well known for his outstanding blog, which has been up for over 15 years without a glitch. If you ask me, keeping up with highest quality for such a long time in an online site is such a huge achievement that even if Peter had hit a pot of gold with his recent attempt at showing how to create the basis of a unification theory based on twistor space, I would still say that comes second as an achievement. But I realize I am getting a bit too fast forward here, so I will back up a little.
So, first of all, what is a twistor? A twistor, like a vector, is better defined not as and of itself, but as a member of a set that contains it: a twistor space. That is because these mathematical objects are defined by their properties, and they inherit their properties by belonging to a space. In a sense, it is the space that has those properties in the first place. And a twistor space is a three-dimensional complex manifold which fulfils some special properties.
Your interest on knowing more of this topic has, I am sure, already declined significantly as we hit mathematical jargon, so I will stop there, pointing the few of you who really would like to know more to the excellent Wikipedia. The relevance to particle physics of twistors comes from the fact that these objects can be constructed from pairs of spinors: since spinors are mathematical objects that describe bits of physical reality (fermions are constructed with spinor fields), the study of twistors might give us some hint of the properties of elementary matter fields. This, of course, is a speculation, and I don't even want to get started on the debate of whether physical reality should be assumed to be based on beautiful mathematics or not - for that, bigger bloggers have written and broadcast so much that I prefer to stay away from the whole thing.
Twistor spaces are the topic of the blog post at Peter Woit's site. There, he explains how he got enthused by their properties, as he saw in particular that they can be brought to have a deep connection with the internal symmetry properties of fundamental fields.
An internal symmetry is a symmetry that is not realized in spacetime, but in some other "internal space" that describes the properties of the object. Electric charge, for instance, is a quantity that describes the symmetry properties of particles withstanding electromagnetic interactions. E.g., if you change all electric charges of a system of particles, whatever that is, by switching positive ones to negative ones and vice-versa, you get a system that behaves exactly in the same way.
Because fundamental fields (the particles the world is made of, and the particles that propagate the interactions between them, and ultimately keep our world together) possess several distinct internal symmetries, it goes without saying that any mathematical structure that can describe the transformation properties of the connected attributes of the fields should get our attention, if we are about trying to decypher the inner fabric of our universe.
In his post Woit describes much better than I could do here why twistors might be a key to the construction of a theory of everything, so I suggest that you visit his post if you want to know more. If you are happy with my sound bytes, the most intriguing thing is that the twistor space "contains" both the Minkowsky space of space-time transformations and the Euclidean space as "slices" of its complex structure.
Now, we have learned long ago, from seminal papers by Glashow, Weinberg and Salam, how the structure of fundamental particles and interactions, described by the standard model, results from the decomposition of a larger group into subgroups that describe the relevant physical properties of those particles. Ever since we did, theorists have tried to investigate what larger mathematical group structure could best contain those physically manifest groups as subgroups, in a way that would both explain the observed physics, but also incorporate all fundamental interactions in a single mathematical structure. That, the "unification dream", has since then been the holy grail of theoretical physicists.
In short, I find it very commendable how Woit is breaking the rule that wants physicists to give important theoretical contributions only at a young age: he is still young-looking, true, but likely he does not pass as "young" by ID card check anymore. And I would be happy to know that somebody can pick up his idea and try to construct some predictive theory out of it.