My book defines natural projection as such: Let $S$ be a set and let ~ be a equivalence relation on $S$. The function $\pi(x)=[x]$ for all $x\in S$ is called the natural projection from $S$ onto the factor set
$S$\~. My book defines $S$\ $f$ as the factor set determined by $f$ where $f:S\rightarrow T$ as the collection of all the equivalence classes of S under ~$_f$. The factor set I understand its just another name for the partition of all the equivalence classes of S brought about the function. But what I don't get is the what the natural projection function does? It seems like it takes each element of set $S$ and matches it up to the appropriate equivalence class in the factor set. I'm not sure if my way of thinking about it is correct?. This is from Beachy and Blair's abstract algebra book. Here is a picture of the composition mapping
It seems your understanding is perfectly fine. The natural projection $\pi$ is something that is uniquely associated to an equivalence relation $~$ on a set $S$. Given such a set and an equivalence relation on it, the projection function associated to it is $\pi:S\to S/\sim$ and it maps, as you say, the element $x\in S$ to its equivalence class $[x]$. It is immediate that the natural projection is surjective, but typically not injective. Another common terminology for this natural projection is the canonical projection. Its of immense importance throughout mathematics.