Intuition behind definition of spinor Some time ago I searched for the definition of spinors and found the wikipedia page on the subject. Although highly detailed the page tries to talk about many different constructions and IMHO doesn't give the intuition behind any of them.
As far as I know physicists prefer to define spinors based on transformation laws (as with vectors and tensors), but all due respect, I find these kind of definitions quite unpleasant. Vectors and tensors can be defined in much more intuitive ways and I believe the same happens with spinors.
In that case, how does one really define spinors without resorting to transformation properties and what is the underlying intuition behind the definition? How the definition relates to the idea of spin from Quantum Mechanics?
In Wikipedia's page we have two definitions. One based on spin groups and another based on Clifford Algebras. I couldn't understand the intuition behind neither of them, so I'd like really to get not just the definition but the intuition behind it.
 A: This is a quite deep and complex topic, which certainly needs a several pages article for a decent intro into it. 
Spinors (although informally, they were already in use by the end of the $19^{th}$ century) are attributed to Elie Cartan. 
Intuitively (not formally), one can say that they "look" like a kind of generalization of the Euler angles: in the sense that they are used to parameterize and describe generalized rotations (in generalized spaces) in a way reminiscent to the use of the Euler angles in the parameterization of $3d$ rotations.
Cartan's initial idea involved the abstract desription of rotations of $3d$ complex vectors: We consider the complex vector space $\mathbb{C}^3$ "equipped" with the product:
$$\mathbf{x}\cdot\mathbf{y}=x_1y_1+x_2y_2+x_3y_3$$
whith $\mathbf{x}=(x_1,x_2,x_3),\mathbf{y}=(y_1,y_2,y_3)\in\mathbb{C}^3$. Then we consider the set of "isotropic" (i.e.: orthogonal to themselves) vectors characterized by 
$$\mathbf{x}\cdot\mathbf{x}=0$$
The set of isotropic vectors of $\mathbb{C}^3$ can be shown to form a $2d$ "hypersurface" inside $\mathbb{C}^3$ and this hypersurface can be parameterized by two complex coordinates $u_0$, $u_1$:
$$\begin{array}{c}
u_0=\sqrt{\frac{x_1-ix_2}{2}} \\ 
u_1= i\sqrt{\frac{x_1+ix_2}{2}}
\end{array} \ \ \ \
\textrm{or} \ \ \ \ 
\begin{array}{c}
u_0=-\sqrt{\frac{x_1-ix_2}{2}} \\ 
u_1=- i\sqrt{\frac{x_1+ix_2}{2}}
\end{array}
$$
Cartan used the term spinor for the complex $2d$ vectors $\mathbf{u}=(u_0,u_1)$. From this, the original isotropic vector $\mathbf{x}=(x_1,x_2,x_3)$ can be easily recovered. He then proceeded to describing the rotations of $\mathbf{x}$ in terms of the rotations of $\mathbf{u}$. 
For a more modern ... "skratch" on the ... "surface" of these ideas, the notes:
http://ocw.mit.edu/resources/res-8-001-applied-geometric-algebra-spring-2009/lecture-notes-contents/Ch5.pdf might prove useful.
A classic -and according to my opinion, invaluable- source is the work of Claude Chevalley: "The algebraic theory of Spinors and Clifford algebras", Collected works, v.2, Springer, 1995. The classic point of view (spinors as generalized complex spaces upon which the Pauli matrices and more generally Clifford algebras act) is further analyzed. Some useful references (up to my opinion) can also be found at: 
https://hal.archives-ouvertes.fr/hal-00502337/document
http://www.fuw.edu.pl/~amt/amt2.pdf
http://cds.cern.ch/record/340609/files/9712113.pdf
http://hitoshi.berkeley.edu/230A/clifford.pdf
Regarding the intuition thing about spinors. Maybe it would be useful at this point to recall that in Classical physics the description is based upon a "rigid" euclidean $3d$ background i.e. vector spaces and euclidean geometry, upon which calculus is performed and produces the quantitative prediction (which is to be tested against experiment). On the other hand, when quantum mechanics and "quantization" comes into play (in almost all elementary senses of the word quantization), the description of the states of a system is based on vectors living inside Hilbert spaces -often infinite dimensional- upon which algebras of "observables" act. The quantitative predictions are now probabilistic and consist of  "spectrums" of eigenvalues of the observables. 
When coming to the description of the problem of rotations, the classical physics recipe consists of using the euler angles as parameters i.e. as a kind of $3d$ coordinates leading thus to orthogonal and generalized orthogonal Lie groups. In the quantum picture, the "parameters" are now special vectors of quotient spaces of hilbert spaces, i.e. "spinors", upon which rotations, which are now for example, elements of Lie groups, Lie algebras, Pauli matrices, elements of Clifford algebras etc, act. 
A: Rotations in three-dimensional space can be represented by the usual real 3 x 3 matrices. They work on real 3 x 1 column matrices which represent a vector. By doing so they yield the representation of the rotated vector. But rotations can also be represented by complex 2 x 2 matrices working on complex 2 x 1 column matrices. This is the representation SU(2). The complex 2 x 1 column matrices do here no longer represent vectors but rotations. They contain the information about the Euler angles, but also (by equivalence) the information about the rotation axis and the rotation angle, or the information about the image of the triad of basis vectors under the rotation. In fact, one can consider them as a steno for the 2 x 2 matrices by writing only their first column because the second column is unambiguously defined by the first column, and this property is preserved under multiplication with SU(2) matrices. In fact the matrices of SU(2) are of the form:
$ a \quad -b^{*}$
$b \quad \quad a^{*}$
The spinor is just the first column of that matrix.
The 2 x 1 spinor matrices are normalized to 1, in conformity with the definition of SU(2): $aa^{*} + bb^{*} =1$. They therefore contain the equivalent of three independent real parameters, i.e. the three Euler angles, or the unit vector along the rotation axis plus the the rotation angle, etc...
The question remains how we put the information about the rotation into these 2 x 2 matrices. That is done by writing the rotations as a product of reflections. (It is easy to verify that the product of two reflections is a rotation. The intersection of the reflection planes is the axis of the rotation. The angle of the rotation is twice the angle between the reflection planes. The reflections are thus generating the group of rotations, reflections and reversals). 
Such a reflection matrix ${\mathbf{A}}$ is easy to find. One uses the unit vector ${\mathbf{a}}$ perpendicular to the reflection plane. The coordinates $a_x, a_y, a_z$ of ${\mathbf{a}}$ must occur somehow in the reflection matrix but we do not know how. Therefore we write the matrix heuristically as $a_x {\mathbf{M}}_x + a_y {\mathbf{M}}_y + a_z {\mathbf{M}}_z$. The matrix ${\mathbf{M}}_x$ will tell where and which coefficient $a_x$ will occur in ${\mathbf{A}}$. Analogous statements apply for the matrices ${\mathbf{M}}_y$ and ${\mathbf{M}}_z$. E.g. if the matrix ${\mathbf{M}}_z$ is
$ 1 \quad \quad 0$
$0 \quad -1$
then the matrix ${\mathbf{A}}$ will contain $a_z$ in position (1,1) and $-a_z$ in position (2,2). The same is true, mutatis mutandis, for the matrices ${\mathbf{M}}_x$ and ${\mathbf{M}}_y$. To find the expressions for these three matrices we express that a reflection is its own inverse, i.e. ${\mathbf{A}}^{2}=1 $ where $1$ stands for the unit matrix. This condition can be satisfied if the matrices ${\mathbf{M}}_j$ satisfy the conditions ${\mathbf{M}}_{x} {\mathbf{M}}_{y} + {\mathbf{M}}_{y} {\mathbf{M}}_{x} = {\mathbf{0}}$ (cycl.) and ${\mathbf{M}}_{x}^{2} = 1$, ${\mathbf{M}}_{y}^{2} =1$,  ${\mathbf{M}}_{z}^{2} =1$. In other words, the matrices ${\mathbf{M}}_j$ can be just taken to be the Pauli matrices $\sigma_{x}, \sigma_{y}, \sigma_{z}$. The matrix becomes then:
$ \quad a_z \quad\quad\quad a_{x} -\imath a_{y}$
$a_{x} + \imath a_{y}  \quad \quad -a_{z}$
Once we have the reflection matrices, we can obtain the rotation matrices by multiplication. This leads to the Rodrigues formula. The spinors are just the first columns of these rotation matrices. If you want it in more detail (and exactly along the lines explained here), you can read it in the third chapter of  "From Spinors to Quantum Mechanics" by G. Coddens (Imperial College Press). You will there also find the link with isotropic vectors mentioned by KonKan, and with the stereographic projection. 
This shows that a spinor is a way to write a group element. That idea remains valid when you want to develop the group representation theory for spinors of the homogeneous Lorentz group in Minkowski space-time (with a few complications, among others due to the metric; Instead of three Pauli matrices you will now need four 4 x 4  gamma matrices. If you know the Dirac theory, you will recognize that this procedure is exactly the way Dirac obtained the gamma matrices. But he used the energy-momentum four-vector $(E,c{\mathbf{p}})$ to define the basis rather than the four unit vectors ${\mathbf{e}}_{\mu}$ of space-time. It is all explained in detail in the reference above).  
As there is a 1-1-correspondence between a set of basis vectors and the rotation that has produced it by operating on the canonical basis, you can thus visualize the spinor as a rotated basis. This remains true in Minkowski space time (now with a set of four basis vectors and with Lorentz transformations). In ${\mathbb{R}}^{3}$ we have ${\mathbf{e}}_{z} = 
{\mathbf{e}}_{x} \wedge {\mathbf{e}}_{y}$, such that ${\mathbf{e}}_{x}$ and ${\mathbf{e}}_{y}$ are actually sufficient to reproduce the whole information. You can therefore also represent the complete information by the isotropic vector ${\mathbf{e}}_{x} +\imath {\mathbf{e}}_{y}$. After rotating ${\mathbf{e}}_{x} +\imath {\mathbf{e}}_{y}$ you can just find 
${\mathbf{e}}'_{x}$ and ${\mathbf{e}}'_{y}$ by taking the real and imaginary parts of the rotated isotropic vector ${\mathbf{e}}'_{x} +\imath {\mathbf{e}}'_{y}$, and thus reconstruct the full rotated basis. This is the reason why one can also represent the rotations by using isotropic vectors. The rotated isotropic vectors are in 1-1-correspondence with the rotations.
It is not any more difficult than that. It is simple geometry. There should not remain any secret or mystery that stands in your way to fully understand this. It is never well explained, which is probably due to the extremely concise  way Cartan introduced them, without explaining what is going on behind the scenes. I hope this helps. Kind regards.
