Given the free group $F_S$, we can define a normal subgroup of $F_S$ as follows. For a given element $w \in F_S$ and $s \in S$, define "the count of $s$", $c_s(w)$, as the number of times $s$ appears in $w$ as written as a word. Then the subgroup of $F_S$ defined as $$N_s = \{w \in F_S \;|\; c_s(w)-c_{s^{-1}}(w) = 0 \}$$ is normal in $F_S$.
This normal subgroup must have a name. What is it? I am unfamiliar with group presentations and couldn't find one.
This is exactly the normal subgroup generated by $S\setminus\{s\}$. Clearly $N_s$ contains $S\setminus\{s\}$. Conversely, if $N$ is any normal subgroup that contains $S\setminus\{s\}$, then any element of $N_s$ maps to the identity in the quotient $F_S/N$ (since all the elements of $S$ except for $s$ map to the identity and so can be removed from the word), so $N_s\subseteq N$.