What do the four components of Dirac Spinors represent in the Standard Model? I've been trying to get my head around the formalisms used in the Standard Model. From what i've gathered Dirac Spinors are 4 component objects designed to be operated on by Lorentz Transformations much like 4-Vectors are in Special Relativity. However they also incorporate additional information: Spin and "Handedness". Due to the nature of Spin, Spinors also transform differently then vectors.
This leaves me with the impression that the 4 components can be classified as: Left Handed and  Spin Up, Left Handed and Spin Down, Right Handed and Spin up, Right Handed and Spin Down.
My question is if this impression is the right general idea or not?
 A: 
the 4 components can be classified as: Left Handed and Spin Up, Left Handed and Spin Down, Right Handed and Spin up, Right Handed and Spin Down.

Your classification is representation-dependent in the context you put it.
However, there is a representation-independent way of doing the same with two (orthogonal) projections:
$$\begin{aligned}
\text{Chiral Projection}&:\ \frac{1}{2}(1\pm \gamma^5) = \frac{1}{2}(1\pm i\gamma^0\gamma^1\gamma^2\gamma^3), \\
\text{Spin Projection}&:\ \frac{1}{2}(1\pm 2S^3) = \frac{1}{2}(1\pm i\gamma^1\gamma^2).
\end{aligned}$$
The specific representation of yours is to choose the 4 eigenvectors of above projections as the base. 
A: It depends on the representation of the gamma-matrices. It can be, for example, the four combinations of electron/positron and spin up/spin down. 
A: In a case of representations $\left( \frac{m}{2}, \frac{n}{2}\right)$ of the Lorentz group we need to take the direct sum of $\left( \frac{m}{2}, \frac{n}{2}\right) + \left( \frac{n}{2}, \frac{m}{2}\right)$, if we want to make our representations irreducible. It is caused by acting on discrete space (and time) inverse operators on space of the Lorentz group: they transfer $\left( \frac{m}{2}, \frac{n}{2}\right)$ representation to $\left( \frac{m}{2}, \frac{n}{2}\right)$, so $\left( \frac{m}{2}, \frac{n}{2}\right)$ alone is not the representation of the full Lorentz group. Also, if we want to make our field real (not complex), we also must to take the direct sum of reps (by analogical reasoning). But then we must to act on direct-sum-rep field by projection operator, which leave only $n + m + 1$ independent components of a field as it must be for spin $s = \frac{n + m}{2}$ field. 
So, let's talk about special case. Dirac bispinor refers to the direct sum of $\left(\frac{1}{2}, 0\right)$ and $\left( 0, \frac{1}{2}\right)$ representations, which correspond to left-handled and right-handled representations (calling chirality). Each of this representations refer to spin $\frac{1}{2}$-particle, and it's projection can be $\pm \frac{1}{2}$. But Dirac equation, which is the projection operator on two-dimensional space of independent components (as it must be for spin $\frac{1}{2}$ be), mixes these components in general. however in a case of $m = 0$ components of different chirality isn't mixed between each other, and Dirac equation leads to two independent equations which are called Weyl's equations. This may be even in Dirac basis. 
Also the anticommutation relations between Dirac matrices and the form of the Dirac equation don't change under unitary transformations: $\gamma ' = U\gamma U^{-1}, \Psi ' = U\Psi$, so by taking $U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 && \sigma_{y} \\ \sigma_{y} && -1\end{pmatrix}$ you make spinor's equations independent. So in this case you also may use your classification.
A: A two component spinor can be geometrically interpreted as representing a point on the Riemann sphere, defined by the ratio of its two complex components, and its stereographic projection onto the xy-plane. Similarly, a four component spinor can be interpreted, by a more complicated ratio defined by its four complex components, as a point on the Riemann sphere followed by a Lorentz transformation, and its stereographic projection onto the complex projective plane $P^2_C$. My recent work, "Vector Analysis of Spinors", and "Spacetime Algebra of Dirac Spinors", addressing these perplexing issues can be found on my website: http://www.garretstar.com/
