Currying

Transforming a function in such a way that it only takes a single argument

Uncurrying is the reverse transformation, and is most easily understood in terms of its right adjoint, the

That is, the parenthesis are not required to disambiguate the order of the application.

The practical motivation for partial application is that very often the functions obtained by supplying some but not all of the arguments to a function are useful; for example, many languages have a function or operator similar to plus_one. Partial application makes it easy to define these functions, for example by creating a function that represents the addition operator with 1 bound as its first argument.

So, a partial application can be defined as the objective result of a single application of the curry operator on some ordering of the inputs of some function.