An n-ary relation is a subset of the cartesian product of $n$ sets. Most common are binary relations. Given a set $A$ and a set $B$ we can form a binary relation $R$ as a subset of $A \times B$. For example if we have $$ A = \{1,2,3\} $$ and $$ B = \{a,b\} $$ we can form the following relations $$ R_1 = \{ (1,a),(2,a) \} $$ or $$ R_2 = \{ (1,a),(2,b),(3,a) \} $$ However the relation $$ R_{not} = \{ (1,1),(a,1) \} $$ is not a subset of $A \times B$ since $(1,1) \not \in A \times B$ and $(a,1) \not \in A \times B$.
For a relation, $R$, we tend to describe that two elements, $a,b$, are related by $a\ R\ b$ - meaning $a$ is related to $b$ by relation $R$. The order is important. $a\ R\ b$ doesn't necessarily imply that $b\ R\ a$.