A coordinate system is a mathematical space, where the elements of the space are called positions. Each position is described by a list of numbers. The length of the list corresponds to the dimension of the coordinate system. So in a 2D coordinate system each position is described by a list containing 2 numbers.
However, in a coordinate system, not all lists of numbers correspond to a position - some lists may be outside the domain of the coordinate system. For example, in a 2D Lat/Lon coordinate system, the list (91,91) does not correspond to a position.
Some coordinate systems also have a mapping from the mathematical space into locations in the real world. So in a Lat/Lon coordinate system, the mathematical position (lat, long) corresponds to a location on the surface of the Earth. This mapping from the mathematical space into real-world locations is called a Datum.