We are constantly thinking of the elements of $\R ^2$ as points of a plane.1
We commonly associate elements of $\R ^2$ with points on a plane. (see Geometry).
To visualize the correspondence we draw two perpendicular lines. We then associate a point of the line with $(0, 0) \in \R ^2$. We can label it so. We then pick a unit length. And proceed as usual.2
Given that we have identified a plane with $\R ^2$ in this way, we call $(x, y) \in \R ^2$ the coordinates of the point it corresponds to. Many authors refer to this identification as a Cartesian coordinate system (or Rectangular coordinate system, $x$-$y$ coordinate system).