a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space.(Wikipedia)

A coordinate change is a set of functions $f^r(x^1, ..., x^N)$ $(r = 1, ..., N)$.

It assigns a new set of coordinates $(x'^1, ..., x'^N)$ to a point $(x^1, ..., x^N)$, wit $x'^r = f^r(x^1, ..., x^N)$ $(r = 1, ..., N)$.