As you know that zeeman splitting is due to the phenomena known as spatial quantization. i.e. if there is a fixed or preferred direction in the space (i.e. symmetry of the space is broken by the electric or magnetic field) then the atom can not assume arbitrary orientation. This orientation depends on the angular momentum of the atomic/spectroscopic state. For example the $^2S$ spectroscopic levels have $0$ orbital angular momentum and $\frac{1}{2}$ spin angular momentum hence it can assume only two states (corresponding to magnetic quantum number $m=-\frac{1}{2}$ and $+\frac{1}{2}$.
There can be two types of coupling i.e. LS coupling and jj coupling. These coupling arises due multi-electronic interaction and tells us how the spin and orbital angular momentum of different electrons are interacting with each other. In LS coupling the orbital angular momentum of the different electrons interact with each other to give the final angular momenta and the spin angular momentum will interact with each other to give final spin momenta. Then the final angular momentum is obtained from these final orbital and spin momenta.
In the case of two electrons with momenta $(l_1,s_1)$ and $(l_2,s_2)$ (of course $s_i$ are just $\frac{1}{2})$
L= $|l_1-l_2|$ to $|l_1+l_2|$
S=$|s_1-s_2|$ to$|s_1+s_2|$
Final momentum
J=$|L-S|$ to $|L+S|$
similarly in jj coupling first the $L_i$ and $s_i$ combine to make $j_i$ and then these $j_i$ combine to make final $J$.
in a magnetic field an atom can attain $m$ spatial states where
$m=-J$ to $+J$.
Hence you will see splitting in the spectral lines, this splitting increases with the increase in magnetic field.
It may be noted here that if the magnetic field is very strong the LS or jj coupling might break and you will see change in splitting. This is known as Paschen-Back effect.
I would like to add here that the terms written in your question $^2P_{3/2}$ are known as spectroscopic terms and calculated on the basis of LS coupling.
For entry level knowledge please consult the book "Introduction to Atomic Spectra" by H.E. White. You may find some portions of this book little difficult. The zeeman effect is described in chapter X. To understand that I suggest you to read earlier chapters.
The understanding of the atomic spectra is a little laborious job and take some effort.
EDIT: To explain sodium D lines
The sodium D lines are from transitions $3^2P_{3/2}\rightarrow3^2S_{1/2}$ ($D_2$) and $3^2P_{1/2}\rightarrow3^2S_{1/2}$ ($D_1$)
Now under the action of magnetic field the upper level of $D_2$ line is split into 4 levels corresponding to m$=\frac{3}{2},\frac{1}{2},-\frac{1}{2},-\frac{3}{2}$ and lower level is split into two levels corresponding to m$=\frac{1}{2},-\frac{1}{2}$. The selection rule require $\Delta$m=$\pm1,0$
and hence you will see transition from m=$\frac{3}{2}\rightarrow\frac{1}{2},\frac{1}{2}\rightarrow\frac{1}{2},\frac{1}{2}\rightarrow-\frac{1}{2},-\frac{1}{2}\rightarrow\frac{1}{2},-\frac{1}{2}\rightarrow-\frac{1}{2}$ and $-\frac{3}{2}\rightarrow-\frac{1}{2}$, simmilarly you can calculate 4 components of $D_1$ line.
I hope this will help