Give me a "hands on"/application oriented definition of what the fuck a spinor is. I know there's a bunch of shit involving spin representations of double covers of SO(n) but none of that shit is concrete. I can't do anything with it. It doesn't tell me how to calculate things involving spinors.

For example: a (p,q) tensor is just something that eats p vectors and spits out q vectors such that it's linear in every argument. Since it's a linear map I can just choose a basis, and then I have coefficients of my tensor and can add them, scalar multiply them, feed them vectors and do shit with them. Give me something like this for spinors.