Modified Spectroscopic Notation (Shankar)

Let's take the same example as given in the book $${}^{2S+1}L_J={}^{2}P_{3/2}$$ This implies $j=3/2$ and letter $P$ denote that $l=1$ from here it's trivial to see $s=j-l=1/2$ which can be seen from the fact that $2s+1=2Rightarrow s=1/2$.

The $S$ is the total spin which might not be a statistic of the particles for a multiparticle system. For example the second example $${}^{2S+1}L_J={}^{1}S_{0}$$

$SRightarrow l=0$ (total orbital angular momentum) and $j=0$ total angular momentum and $2s+1=1rightarrow s=0$ that is total spin angular momentum which doesn't denote the statistics of the particles.

$2S+1$ is the multiplicity, meaning how many degenerate states there are for a given $S$ since $-S leq M_s leq S$ (where $M_s$ is the spin projection number). For $S=1$, $M_s = -1, 0, 1$. So the superscript is $2(1) + 1 = 3$ for the $3$ degenerate states in spin.