Confusion between operation and relation: Clarification needed

A binary operation is a function from $S\times S \to S$ such as addition, multiplication or anything really.

A binary relation is just a subset of $S^2$, that is not necessarily a function and it doesn't have to include all the elements of $S$ in one way or another.

A function $f\colon S\to R$ is a relation, this time it's a subset of $S\times R$ however it satisfies a certain property, if you take some $s \in S$ then there is only a unique ordered pair with $s$ in it, so if you have $\langle s,r_1\rangle$ as well $\langle s,r_2\rangle$ then you can say that $r_1 = r_2$.