Dirac, Weyl and Majorana Spinors To get to the point - what's the defining differences between them? Alas, my current understanding of a spinor is limited. All I know is that they are used to describe fermions (?), but I'm not sure why? 
Although I should probably grasp the above first, what is the difference between Dirac, Weyl and Majorana spinors? I know that there are similarities (as in overlaps) and that the Dirac spinor is a solution to the Dirac equation etc. But what's their mathematical differences, their purpose and their importance?
(It might be good to note that I'm coming from a string theory perspective. Plus I've exhausted Wikipedia here.)
 A: Recall a Dirac spinor which obeys the Dirac Lagrangian
$$\mathcal{L} = \bar{\psi}(i\gamma^{\mu}\partial_\mu -m)\psi.$$
The Dirac spinor is a four-component spinor, but may be decomposed into a pair of two-component spinors, i.e. we propose
$$\psi = \left( \begin{array}{c} u_+\\ u_-\end{array}\right),$$
and the Dirac Lagrangian becomes,
$$\mathcal{L} = iu_{-}^{\dagger}\sigma^{\mu}\partial_{\mu}u_{-} + iu_{+}^{\dagger}\bar{\sigma}^{\mu}\partial_{\mu}u_{+} -m(u^{\dagger}_{+}u_{-} + u_{-}^{\dagger}u_{+})$$
where $\sigma^{\mu} = (\mathbb{1},\sigma^{i})$ and $\bar{\sigma}^{\mu} = (\mathbb{1},-\sigma^{i})$ where $\sigma^{i}$ are the Pauli matrices and $i=1,..,3.$ The two-component spinors $u_{+}$ and $u_{-}$ are called Weyl or chiral spinors. In the limit $m\to 0$, a fermion can be described by a single Weyl spinor, satisfying e.g.
$$i\bar{\sigma}^{\mu}\partial_{\mu}u_{+}=0.$$
Majorana fermions are similar to Weyl fermions; they also have two-components. But  they must satisfy a reality condition and they must be invariant under charge conjugation. When you expand a Majorana fermion, the Fourier coefficients (or operators upon canonical quantization) are real. In other words, a Majorana fermion $\psi_{M}$ may be written in terms of Weyl spinors as,
$$\psi_M = \left( \begin{array}{c} u_+\\ -i \sigma^2u^\ast_+\end{array}\right).$$
Majorana spinors are used frequently in supersymmetric theories. In the Wess-Zumino model - the simplest SUSY model - a supermultiplet is constructed from a complex scalar, auxiliary pseudo-scalar field, and Majorana spinor precisely because it has two degrees of freedom unlike a Dirac spinor. The action of the theory is simply,
$$S \sim - \int d^4x \left( \frac{1}{2}\partial^\mu \phi^{\ast}\partial_\mu \phi + i \psi^{\dagger}\bar{\sigma}^\mu \partial_\mu \psi + |F|^2 \right)$$
where $F$ is the auxiliary field, whose equations of motion set $F=0$ but is necessary on grounds of consistency due to the degrees of freedom off-shell and on-shell.
A: After you will learn more about spinors, you will see that all spinors belong to the $\left(\frac{1}{2}, 0\right) + \left( 0, \frac{1}{2}\right)$ representation of the $SL(2,C)$ group, which is the double cover of the lorentz group $SO(3,1)$. The idea is to find representations of a simply connected covering group which in this case is $SL(2,C)$, the local structure given by the lie algebraic commutation relation remains the same. 
Spinorial equations allow to extract Lorentz-invariant subspaces in the overall space of $\left(\frac{1}{2}, 0\right) + \left( 0, \frac{1}{2}\right)$ representation.
Both Dirac and Majorana spinors belong to $\left(\frac{1}{2}, 0\right) + \left( 0, \frac{1}{2}\right)$ representation of $SL(2,C)$ group, but they are only subspaces of it. For instance, Majorana spinors are all electrically neutral (i.e. remain invariant under charge conjugation). Similarly, Dirac spinors are "magnetically neutral".
Weil spinors belong to either $\left(\frac{1}{2}, 0\right)$ or $\left( 0, \frac{1}{2}\right)$ subspaces. Unlike Dirac and Majorana spinors, they might be considered as 2-component spinors. But this is also a limitation, because some special Lorentz transformations cannot be applied to these spinors.
