"In Minkowski space, a twistor may be defined as a pair consisting of a spinor field and a complex conjugate spinor field satisfying the twistor equation."
http://mathworld.wolfram.com/Twistor.html
Minkowski Space--"A four-dimensional space with the Minkowski metric. Minkowski space unifies Euclidean three-space plus time (the "fourth dimension") in Einstein's theory of special relativity. "
Minkowski Metric-- click here to see
Spinor Field-- click here to see
Twistor Equation--click here to see
As you can see, dear reader, this is not your average first year stuff. Penrose founded the geometry in the 70s. Since then, it has found a variety of applications. Mainly, Penrose himself is applying it as a way to integrate or at least compromise between the two disparate but fundamental theories of quantum mechanics and general relativity by redefining space-time in terms of twistor theory. The two theories are each amazingly precise, but in their parameters. QM is amazing on the extremely small scales, while GR is incredibly, astonishingly, mindnumbingly accurate on a large scale (10 to the -14 accuracy, I believe). Twistor theory proposes a method of integrating them after a fashion, by redefining both in terms of twistor geometry and leading the respective theories to a convergence, leading us one step closer to a Unified Field Theory.
Twistor Theory is actually the heart of Penrose's 30 years of research, although as one site's author, Andrew Hodges, points out, you'd hardly know it by the public knowledge of his work.
It can additionally be used to address non-linear equations in a mathematical sense
It is also confusing. The primary reason it is rather obscure is because the language of the theory itself is necessarily describing 4 dimensions, and reads like Sanskrit to the undereducated. (meaning non-math grad students).