In Maggiore A Modern Introduction to Quantum Field Theory Eq. 4.31
$$\{\Psi_a(\vec x,t),\Psi_b(\vec x,t)\}=\delta^{(3)}(\vec x-\vec y)) \delta_{ab}$$
where "$a,b=1,2,3,4$ are the Dirac indices".
However, The definition of Dirac indices were nowhere to be find.
I knew that there were four independent basis for the field (for $s=1,2$), did they mean that? i.e. $a=1,2$ for $s=1$ and $a=3,4$ for $s=2$?
What's the Dirac indices?
For the $u$-spinor solutions $u_{a}(s,p)$, which you mention in the comments, the spinor index $a$ runs over $a = \{1,2,3,4\}$, whilst independently $s$ refers to the two different spin states. And we have the $v_{a}(s,p)$ for the negative energy solutions.
Take $\mathbf{p} = (0,0,p_z)$, then the positive energy spinor solutions for the different values of $s=1,2$ are \begin{align} u_{a}(1,p) = \begin{bmatrix} \sqrt{E+m} \\ 0 \\ \sqrt{E-m} \\ 0 \end{bmatrix} \quad \quad u_{a}(2,p) = \begin{bmatrix} 0 \\ \sqrt{E+m} \\ 0 \\ -\sqrt{E+m} \end{bmatrix} \ . \end{align} The spinor index $a$ still runs over $\{1,2,3,4\}$, so for example $u_{1}(1,p)=\sqrt{E+m}$. As you probably know, when you write the fields $\Psi_a (\mathbf{x},t)$ explicitly we include the $u$- and $v$-spinor solutions, whilst summing over $s$, with the spinor index $a$ being free.
If you're asking more generally about the space of the Dirac/spinor indices, I think there's already questions on that here, e.g. spinor vs vector indices of Dirac gamma matrices. There's also quite a lot of resources online about spinors generally, e.g. https://arxiv.org/abs/1312.3824
