Binary Operation

A binary operation on a set $S$ is a function

$\ast: S \times S \to S$.

The critical point is closure: whenever $a$ and $b$ lie in $S$, the product $a \ast b$ must also lie in $S$.

Binary operations may or may not be associative, commutative, or possess identities and inverses. Those are extra properties, not part of the bare definition.

One-line intuition

A binary operation is a rule that combines two elements of the set and stays inside the set.