What are Dirac spinors and why did relativistic quantum mechanics need them? I have a good grasp of the Schrödinger equation and the basics of special relativity But the Dirac equation is alien to me. What are Dirac spinors and why did Dirac use them?
 A: Thomas Fritsch's answer summarizes the history of the Dirac equation. Dirac did find it because he was looking for a first-order relativistic wave equation, which he believed was necessary for relativistic QM.
He was wrong, though. The correct formulation of relativistic quantum mechanics turned out to be quantum field theory, in which integer-spin particles are modeled by the scalar and vector field equations that Dirac believed were unusable. If there were no half-integer-spin particles in nature, we wouldn't need spinors for relativistic quantum theory.
Furthermore, no Dirac spinor fields are known to exist in nature. The spin-½ fields in the Standard Model are Weyl spinors.
The alleged predictions of the Dirac equation are a bit overblown. The Dirac equation certainly doesn't predict the electron to have spin ½, as Thomas Fritsch's answer states. The Dirac equation can model the behavior of any Dirac particles that may happen to exist, but doesn't predict that any do exist, much less that the one named "electron" has those properties. And, as I said, no fundamental Dirac particles are known to exist.
Dirac correctly predicted the positron, but his argument for it, involving the "Dirac sea", was wrong. The argument doesn't work for integer-spin fundamental particles, yet they also have antiparticles. (In fact, his argument taken at face value would imply that fundamental bosons can't exist at all—a possibility that I believe was taken seriously for a while in the early days of relativistic QM.)
The electron $g$-factor doesn't theoretically have to be $2$, although that is the simplest possibility, so it could be called a prediction in the sense of Occam's razor. I don't know much about this, but Wikipedia has a short section on it.
In summary, quantum field theory doesn't need Dirac spinors, and to a certain extent Dirac just got lucky.
A: Others explained Dirac's motivation. One of the answers to your question "What are Dirac spinors" is they are a certain representation of the Lorentz group, i.e., they transform consistently under Lorentz transformations. This is one of the reasons of the success of the Dirac equation.
A: To understand what Dirac spinors are and why we need them,
you need to understand where the Dirac equation comes from.
Dirac in 1928 derived a relativistic wave equation describing a particle
with mass $m$. The heuristic requirements for this equation were:

*

*It should be a linear differential equation first-order in time
and space. (This is unlike Schrödinger's non-relativistic wave
equation which is first-order in time and second-order in space.
And it is also unlike the relativistic Klein-Gordon equation
which is second-order in time and space. However, the Klein-Gordon
equation correctly describes spinless particles only, but
fails for particles with spin, e.g. the electron. So it seemed,
a radically new approach was needed.)

*It should have plane wave solutions
$$\psi(x)\propto e^{i(-Et+\vec{p}\cdot\vec{x})/\hbar}$$
thus satisfying the Planck/Einstein relation $E=\hbar\omega$ and
the de Broglie relation $\vec{p}=\hbar\vec{k}$.

*It should reproduce the relativistic energy-momentum relation:
$$\frac{E^2}{c^2}-\vec{p}^2=m^2c^2$$
Requirements 1 and 2 dictates the equation to have the form
$$i\hbar\sum_{\mu=0}^3 \gamma^\mu \frac{\partial \psi(x)}{\partial x^\mu}
  = mc \psi(x) \tag{1}$$
with some still unknown constants $\gamma^0,\gamma^1, \gamma^2, \gamma^3$.
From requirement 3 these constants need to satisfy the condition
$$\gamma^\mu\gamma^\nu + \gamma^\nu\gamma^\mu=2\eta^{\mu\nu} \tag{2}$$
where $\eta^{\mu\nu}$ is the Minkowski metric
(for details of this reasoning see for example at
Three Derivations of the Dirac Equation, section 7).
Obviously this condition cannot be met with ordinary numbers. The most
simple solution to (2) is a set of $4\times 4$-matrices.
The solution is not unique. An example solution (known as the standard
representation of the $\gamma$ matrices) is this:
$$
\gamma^0=\begin{pmatrix}I & 0 \\ 0 & -I\end{pmatrix},\quad
\gamma^1=\begin{pmatrix}0 & \sigma_x \\ -\sigma_x&0\end{pmatrix},\quad
\gamma^2=\begin{pmatrix}0 & \sigma_y \\ -\sigma_y&0\end{pmatrix},\quad
\gamma^3=\begin{pmatrix}0 & \sigma_z \\ -\sigma_z&0\end{pmatrix}
$$
where $0$ and $I$ are the $2\times 2$ null matrix and unity matrix,
and $\sigma_x,\sigma_y,\sigma_z$ are the Pauli matrices.
Then, in order for equation (1) to make sense, the wave function $\psi(x)$
needs to be a thing with $4$ complex components (which we call a Dirac
spinor).

But so far all this was just a speculative theory. The actual
confirmation of the Dirac equation came when its predictions were
compared to experimental reality.

*

*It predicted the electron to have spin $s=\frac{1}{2}$ and
a gyromagnetic ratio of $g=2$. These facts were already known
before, but lacked a theoretical explanation.

*It predicted positrons (particles like electrons, but with
opposite charge). Positrons were experimentally discovered a few
years after the theoretical prediction.

*It correctly predicted the fine-structure of the spectrum of the
hydrogen atom. This fine-structure was already measured
experimentally. But the Klein-Gordon equation gave a slightly wrong
prediction about this.

