A binary operation on a set $X$ is a map $\ast : X \times X \to X$. Usually, we denote $\ast(x, y)$ by $x\ast y$. For questions about operations in binary arithmetic (base 2), use the tag (binary) instead.
Note: Whether or not a given operation is a binary operation can depend on the set. For example, subtraction is not a binary operation on $\mathbb{N}$ but it is on $\mathbb{Z}$.
There are many objects in abstract algebra which require binary operations as part of their definition. These include: magma, semigroup, monoid, quasigroup, ring, and field.
An $n$-ary operation on $X$ which is a map $\ast : X^n \to X$. A binary operation is the special case $n = 2$.
