Spinors in three dimensions
At last I think I understand what this strange business with spinors is really all about. Everybody knows that it has to do with SU(2) being a double cover of SO(3) of course, but I think I finally comprehend why.
SO(3) is the group of proper rotations in 3D space. For instance, a rotation around the $z$-axis by angle $\theta$ can be represented by the following matrix product:
\[\begin{pmatrix}x'\\y'\\z'\end{pmatrix} = \begin{pmatrix} \cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1 \end{pmatrix}\cdot\begin{pmatrix}x\\y\\z\end{pmatrix}.\]
Yet there's another way to represent the same rotation using matrix algebra. We can arrange the coordinates into a 2×2 complex matrix and perform the following multiplication:
\[\begin{pmatrix} iz'&y'+ix'\\-y'+ix'&-iz' \end{pmatrix}= \begin{pmatrix}e^{i\theta/2}&0\\0&e^{-i\theta/2}\end{pmatrix} \cdot \begin{pmatrix}iz&y+ix\\-y+ix&-iz\end{pmatrix}\cdot \begin{pmatrix}e^{-i\theta/2}&0\\0&e^{i\theta/2}\end{pmatrix}.\]
If you carry out the computation (not too difficult) you'll find that this representation of a spatial rotation is equivalent to the previous representation. And it can be easily generalized to an arbitrary rotation (though the calculations tend to get more tedious). What we find is that the complex matrix used to represent a rotation will be a unitary complex matrix (unitary meaning that when multiplied by its own conjugate transpose, the result is the unit matrix) of determinant +1. These matrices form a group under matrix multiplication: the group is called SU(2). In other words, SU(2) is equivalent to SO(3).
Well, almost. Whereas in the first representation, there's precisely one matrix that corresponds to a particular spatial rotation, in the second representation there are two:
\[\begin{pmatrix}e^{i\theta/2}&0\\0&e^{-i\theta/2}\end{pmatrix}{\rm~~and~~} \begin{pmatrix}-e^{i\theta/2}&0\\0&e^{-i\theta/2}\end{pmatrix} = \begin{pmatrix}e^{i\theta/2+\pi}&0\\0&e^{-i\theta/2-\pi}\end{pmatrix}\]
both correspond to the same rotation, since the minus sign, occurring twice, will cancel. Another way of putting this is that SU(2) is a double cover of SO(3). If $\vec{B}$ is an element of SU(2) and $\vec{V}$ is the 2×2 complex matrix representation of a set of coordinates, the rotation can be written up as $\vec{B}\cdot\vec{V}\cdot\overline{\vec{B}}$.
So far so good. These matrices can be used to represent a rotation in 3D space; i.e., a coordinate transformation that corresponds to a proper rotation of the coordinate system. In which case it is customary to talk about covariant and contravariant quantities, i.e., quantities that transform differently under a coordinate transformation.
What spinors make possible is to define a third type of quantity. One that transforms according to $\vec{B}\cdot\vec{V}$, i.e., "one half" of a rotation if you wish. Or (I think Penrose called it that) the "square root" of a rotation. A quantity like this is called a spinor. The most notorious property of a spinor of course is that when $\theta$ is 360º, the spin matrix is not the identity matrix, so it takes a rotation of 720º in order for a spinor to return to itself.
Needless to say, all this stuff can be generalized to higher dimensions, but not arbitrarily. The correspondence between SU(2) and SO(3) is accidental. In general, the dimensionality of SU($n$) is $n^2-1$, and the dimensionality of SO($m$) is $m(m-1)/2$, so corresponding pairs of $n$ and $m$ can be found by solving the Diophantine equation $2(n^2-1)=m(m-1)$. Solutions include the trivial SU(1) = SO(1) (both groups consisting of the single 1×1 "matrix" (1)), $n=2$ and $m=3$, and $n=4$, $m=6$ (SU(4) and SO(6) are isomorphic, i.e., one is not a double cover of the other). There are also solutions to this equation that do not result in a covering at all.
In other words, these kinds of spinors are a rather special counterpart to 3D rotations. They may, in fact, provide a glimpse into that great unanswered mystery: why is the space we live in 3-dimensional?
So what are spinors for anyway? Quantities that transform like a spinor appear in quantum mechanics: the coefficients in the Dirac-equation form a spinor, for instance, because when you rewrite the equation under a change of coordinates, in order for it to remain invariant, the Dirac-coefficients must transform as spinors do.