Dirac Lagrangian
We want to use our spinor fields $\psi_L$ for $(\frac{1}{2},0)$ and $\psi_R$ for $(0,\frac{1}{2})$ to build a Lorentz invariant Lagrangian.
Taking a naive guess like $$(\psi_R)^\dagger\Box\psi_R + m^2(\psi_R)^\dagger\psi_R,$$ one quickly realizes that this is not invariant under the Lorentz transformations $$\psi_L \to e^{\frac{1}{2}(i\Theta_j\sigma_j - \beta_j\sigma_j)}\psi_L$$ $$\psi_R \to e^{\frac{1}{2}(i\Theta_j\sigma_j+ \beta_j\sigma_j)}\psi_R.$$ However we find that e.g. combinations $\psi^\dagger_L\psi_R$ is Lorentz invariant. We can use this to build a mass term:t
Dirac mass term
The Dirac mass term for Spin $\frac{1}{2}$ particles is $$\mathcal{L}_\text{dirac mass}=m(\psi^\dagger_L\psi_R+\psi^\dagger_R\psi_L).$$
It is Lorentz invariant, where it was chosen to consist of two conjugated terms in order for $\mathcal{L}$ to be real.
We want to find a kinetic term. We note that a term like $$\mathcal{L}=\psi^\dagger_L\Box\psi_R + \psi^\dagger_R\Box\psi_L$$ essentially decomposes into the kinetic term for a couple of scalar fields after plugging in the spinors. If the Lagragian corresponds to a theory with a couple of scalar fields, this is not wat we are looking for. This is similar to the Lagragian for $A_\mu$, which needs to be constructed in a certain way to give the dynamics of a vector field. This means the Lagrangian has to be build in a way, which forces the transformation properties of its constituents. For the scalar field we have the contraction $\partial_\mu A_\mu$ in the Lagrangian and since $\partial_\mu$ is a Lorentz vector, $A_\mu$ also has to transform like a vector for the Lagrangian to be Lorentz invariant.
We find kinetic terms which are Lorentz invariant as $$\psi^\dagger_R\partial_t\psi_R+\psi^\dagger_R\partial_j\sigma_j\psi_R$$ $$\psi^\dagger_L\partial_t\psi_L-\psi^\dagger_L\partial_j\sigma_j\psi_L$$ because the vector $(\psi^\dagger_R\psi_R, \psi^\dagger_R\sigma_j\psi_R)$ transforms like a Lorentz vector.
We can put these terms together using $$\sigma^\mu=(1,\vec{\sigma}),\quad\bar{\sigma}^\mu=(1,-\vec{\sigma})$$
Dirac Lagrangian
The Dirac Lagrangian is the Lagrangian for Spin $\frac{1}{2}$ particles with a Dirac Spinor: $$\psi = \begin{pmatrix}\psi_L\\ \psi_R\end{pmatrix},$$ $$\bar{\psi}=\left(\psi^\dagger_R,;\psi^\dagger_L\right)$$
The Lagrangian for particles with mass $m$ is $$\mathcal{L}=\bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi$$ where the Dirac Matrices or gamma matrices are defined as $$\gamma^\mu = \begin{pmatrix}0 & \sigma^\mu \\ \bar{\sigma}^\mu & 0\end{pmatrix}$$
The equations of motion which follows is the Dirac equation $$~~~(i\cancel{\partial}-m)\psi=0$$ $$\bar{\psi}(-i\overleftarrow{\cancel{\partial}}-m)=0$$
Since the Dirac equation was built out of two Spin $\frac{1}{2}$ representations it describes two Spin $\frac{1}{2}$ particles, the particles and the antiparticles