What is a relation in plain simple English?

A relation is a way of describing when two things are related. Say that $P$ is the set of all points of a plane (such as $\Bbb R^2$, for instance) and that $C$ is the set of all circles in that plane. We can say that a point $p$ is related to a circle $c$ if $p\in c$. So, for instance, $(1,1)$ is related to the circle centered at $(0,0)$ with radius $\sqrt2$, but not to the circle centered at $(1,0)$ with radius $1$.

If we call $R$ to this relation, we can defined it formally as$$p\mathrel Rc\text{ if and only if }p\in c.$$

The formal definition if relation consists, in this case, in defining $R$ as the subset of $\Bbb R^2\times\{\text{circles in }\Bbb R^2\}$ which consists of all pars $(p,c)$ such that $p\in c$. And now $p\mathrel Rc$ is just a shortcut for $(p,c)\in R$.