spinor
A spinor is the mathematical object that holds the full state of a spin-½ particle. For a single spin it is a column of two complex numbers, one attached to the up state and one to the down state. The size of each number sets how likely that outcome is, and the relative phase between them encodes which way in space the spin actually points.
What makes a spinor more than just a list of two numbers is how it changes when you rotate it. Ordinary vectors return to themselves after one full turn of three hundred and sixty degrees. A spinor does not: a single full rotation flips its overall sign, and you have to turn it twice, a full seven hundred and twenty degrees, before it comes back exactly as it was. This is the famous double-valued behaviour that sets spin-½ apart.
Spinors are the natural language for everything built from spin-½ particles, from a lone electron to the wavefunctions of atoms and the quantum fields of particle physics. In quantum computing the spinor is just the state of a single qubit. Learning to read a spinor, two complex numbers carrying probability and phase, is learning to read the most basic quantum system there is.
Two complex numbers a and b set the probabilities and phase; a full turn flips the sign.
A spinor's overall sign and phase are not directly observable; only relative phases and the squared magnitudes show up in measurements. The sign flip after one turn is real but only matters in interference between paths.