What is a spinor? In a youtube video, sir Michael Atiyah mentioned that even after working during the most of his life on spinors, he doesn't know what a spinor is. Now surely that was part of his humorous introduction to the talk there, but still, we work on spinors but how much are we capable of setting its definition in simple terms? My question is in the simplest form that I can think of: What is a spinor?
 A: I watched the video. IMHO Sir Michael was being rather humble and diplomatic here, and there were a few things he didn't mention. But here's a few things he did mention:  electron, geometry, complex (=rotation), geometrical meaning, Hodge harmonic forms from Maxwell's equations, Schrodinger wave equation, relativistic Dirac equation, curved space, harmonic spinors, sphere with a slightly deformed metric, topological significance, scalar curvature, circle action, related to Compton wavelength, twistor. I thought he was going to pull everything together at the end and volunteer an answer to *What is a spinor?" but he didn't. So I'll have a go. Please note that this is opinion rather than an authoritative peer-reviewed answer, but I hope it's of some use anyway:

A spinor is a mathematical representation of a harmonic standing-wave quantum  field "topological structure" or excitation which typically exhibits a spin ½ geometry which in turn can be likened to Dirac's belt or the Möbius strip. Like the Dirac spinor this can be thought of as a "bispinor" in that an essential feature is two orthogonal rotations. Note however that waves and fields do not take the form of a flat twisted strip, so mentally "inflate" the Moebius strip to a dynamical twisting turning ring torus, then to a horn torus, then to a spindle-sphere torus. 

Also note that waves and fields do not feature a surface, so consider this spherically-symmetric object to be merely the central portion of a charged particle such as the electron, wherein the Compton wavelength is twice the circumference. This central portion is akin to "the eye of the storm", wherein the electron itself can be likened to a cyclone, and a positron can be likened to an anticyclone. Note that co-rotating vortices repel, whilst counter-rotating vortices attract. So one might say "spinors are the square root of geometry" because the linear electric force and rotational magnetic force between two charged particles results from the interaction of two electromagnetic fields, which are not totally unlike the gravitomagnetic field. In other words, the forces between the electron and positron are present because each is a dynamical spinor in frame-dragged space. As the electron and positron move closer to one another they "exchange field" such that the short-lived positronium atom has very little in the way of a field, and this is modelled in QED via virtual particles aka field quanta or "chunks of field". Interestingly, the concept of spinors can arguably be traced back to Thomson and Tait who coined the phrase spherical harmonics, and to Maxwell and his theory of molecular vortices.    
