Twistor theory is something like incidence geometry: one element of the projective twistor space corresponds to a “line” in the spacetime, whereas one spacetime point (event) corresponds to its “sky”, a submanifold in the projective twistor space. The most simple example is the complexified Minkowski space where projective twistor space is 3-dimensional projective space ℂP3. Projective twistors that correspond to real light lines form a 5-dimensional (real) quadric in ℂP3; they are called projective null twistors.
There is also a purely differentially-geometric way to define the manifold of projective null twistors for any globally hyperbolic Lorentzian spacetime; it doesn’t use complex numbers.
Twistor theory (a.k.a. twistor programme) was developed by British mathematician Roger Penrose and associated researchers. It is used in general-relativity and quantum theories, especially on a curved spacetime.