A binary function, or function of two variables, is like a function but with two domains. That is, while a function $f$ from the set $A$ to the set $C$ maps an element $x$ of $A$ to a unique element $f(x)$ of $C$, a binary function from $A$ and $B$ to $C$ maps an element $x$ of $A$ and an element $y$ of $B$ to a unique element $f(x,y)$ of $C$.
We can generalise further to multiary functions, or functions of several variables.
The possible definitions depend on foundations; for us, the simplest is probably this:
A binary function from $A$ and $B$ to $C$ is simply a function $f$ to $C$ from the binary cartesian product $A \times B$. We write $f(x,y)$ for $f((x,y))$.
This is natural in structural set theory; it makes sense in any set theory and (suitably interpreted) in a foundational type theory.
In material set theory, one often declines to specify the domain of a function (since this can be recovered from it); then we can say this:
A binary function to $C$ is simply a function $f$ to $C$ such that every element of the domain of $f$ is an ordered pair. Again, we write $f(x,y)$ for $f((x,y))$.
An equivalent (but different) definition in material set theory is this:
A binary function to $C$ is a ternary relation $f$ such that
Now we write $f(x,y)$ for the unique $z$ (if any) such that $(x,y,z) \in f$.
Either way, the material concept is actually more general than the structural one, because the domain of a material binary function might not be a cartesian product. The link is provided by the binary version of a partial function:
A partial binary function from $A$ and $B$ to $C$ is a function to $C$ from a subset of $A \times B$.
It's also possible to take the concept of binary function as an undefined primitive concept, on the same level as that of function. Then we want axioms such as the following (depending on the style of foundations):
Given an element $x$ of the set $A$, an element $y$ of $B$, and a binary function from $A$ and $B$ to $C$, we have an element $f(x,y)$ of $C$.
If $x$ and $x'$ are equal elements of $A$ and $y$ and $y'$ are equal elements of $B$, then $f(x,y) = f(x',y')$.
Given two such binary functions $f$ and $f'$, if $f(x,y) = f'(x,y)$ for every $x$ in $A$ and every $y$ in $B$, then $f = f'$.
We could conceivably have the notion of binary function without the notion of cartesian product; then a binary function could not be understood as a special case of a function. I doubt that anybody has proposed such a foundation of mathematics, but there are situations where this is true in some internal logic.
In particular, a binary function internal to a multicategory is simply a binary morphism in that multicategory. This is most like a binary function between sets in the case of a cartesian multicategory. But even so, there may be no tensor product in the multicategory, and then a binary morphism cannot be understood as a special case of a morphism.
binary function, bilinear map, multilinear map