Def: A relation from $S$ to $T$ is a subset of $S \times T$
Statement: A membership relation $\in$ from $S$ to Power Set $P(S)$ is perhaps the most important and basic relationship in mathematics
$S = S, T = P(S)$? What are the elements of relation subset here?
You're right. The elements of the relation subset are $(x, A)$ for each $A \subseteq S$ and each $x \in A$.
If we let $R$ be the "membership relation" subset of $S \times \mathcal{P}(S)$, then we may write $s \in x$ as a shorthand for $(s, x) \in R$.
More generally, if $R$ is a relation (defined as a subset of $S \times T$), then we may write $x \sim y$ as a shorthand for $(x,y) \in R$, and we may subsequently refer to $R$ by the name of $\sim$. (In this instance, $\in$ is a special symbol we use when we know that the relation is "membership".)