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.
As you probably know, there are different irreducible representations of the different symmetry groups that one gets in relativistic quantum field theory. Here we are dealing with the Lorentz symmetry (part of the Poincare symmetry).
One representation, the spin-half representation, is associated with fermions as one finds in the Dirac equation. The indices of this representations are the spin indices, as denoted by the rows and columns of the Dirac matrices. In this representation the Lorentz transformations are represented by spin transformations.
Another representation, the spin-one representation, is associated with vector fields, such as the gauge bosons. It is also the representation for the coordinate vectors. The indices in this case are the space-time indices that are usually denoted by some Greek letter such as $\mu$. In this representation the Lorentz transformations transform the space-time indices (rotations and boosts).
One can combine a pair of spin-half quantities in a why that they act like spin-one quantities. (This is part of a more general process whereby tensor products of irreducible representations of a symmetry group become direct sums of different irreducible representations of that symmetry group.) This means that a product of two spin-half quantities, each transforming by spin transformations, can act like a single spin-one quantity, transforming as a space-time vector, provided that the two quantities are combined in a particular way. It turns out that the Dirac matrices provide the correct way to combine two spin-half quantities so that they transform as a single spin-one quantity. Equation (3.29) demonstrates this property.
Hope this addresses all your questions.
Best Answer
The spin of a particle is its intrinsic angular momentum, a vector quantity unrelated to any actual rotation of the particle. As you learned in quantum mechanics its magnitude is quantized as $\sqrt{s(s+1)}\hbar$ and any of its three components as $m_s\hbar$. Here $s=0,1/2,1,3/2,2,...$ is the principal spin quantum number and $m_s$, which ranges from $+s$ to $-s$ in steps of 1, is the secondary spin quantum number.
When talking about a particle or field, often the value of $s$ is called “the spin”. But sometimes, if one is focused on, say, the $z$-component of the angular momentum, $m_s$ is called “the spin”; it should really be called “the $z$-component of the spin”, but that gets tedious.
So “the spin” can mean (1) the intrinsic angular momentum; (2) the quantum number $s$ that specifies the magnitude of that angular momentum; (3) a component (usually the $z$-component) of that angular momentum; (4) the quantum number $m_s$ that specifies that component.
In quantum field theory, the quanta of various kinds of fields have various spins. A scalar field is said to be “spin 0” because its quanta have $s=0$. A spinor field is said to be “spin 1/2“ because its quanta have $s=1/2$. A vector field is said to he “spin 1“ because its quanta have $s=1$. A tensor field with two indices is said to be “spin 2” because its quanta have $s=2$.
In QFT, you can also think about spin more abstractly in terms of how the field transforms under spatial rotations rather than how much intrinsic angular momentum its quanta have. This connection should not be too surprising, because the conservation of angular momentum is related to invariance under rotations.
Under rotations of the coordinate system, the fields have to transform according to a “representation” of the rotation group. The relevant mathematics is the theory of Lie groups (like the rotation group $SO(3)$ and the larger Lorentz and Poincaré groups) and their representations. A representation is a set of linear transformations in an abstract vector space (often a complex one) of arbitrary dimension that compose in the same way that the abstract group elements do. (“In the same way” means their composition is homomorphic to the group composition.)
Scalar, spinor, vector, and tensor fields are different representations of the same rotation group, in different dimensions. Here the dimensions are not physical dimensions like height and width but “field dimensions”. For example, a Dirac spinor has four components. Under rotation, they mix together linearly, similarly to how $x$, $y$, and $z$ spatial coordinates mix together linearly under a rotation. This four-dimensional complex vector space is an abstract representation space, not a geometric space like spacetime. Weyl and Majorana spinors have two components and “live” in a two-dimensional abstract representation space.
The word “spinor” is used to mean several different but closely related things: (1) A particular representation with $s=1/2$; (2) An element in the representation space, i.e., a multi-component field that transforms according to this representation; (3) A particle that is a quantum of such a field.
A fuller discussion of spinors would get into projective representations, covering groups, and other arcana of group theory. Hopefully this intro will get you oriented to the simpler ideas first.