A coordinate system is a description of the origin (datum) of your map, the projection system used, and the units you are using. It describes how the sphere of the Earth is projected onto a sheet of paper and converted to the Cartesian coordinate system.
Mapping applications use coordinate systems to determine where to display features on the map. In a latitude/longitude coordinate system, each coordinate pair corresponds to an exact fixed location on the Earth; in an arbitrary XY system, the coordinates have meaning in relation to each other but do not necessarily correspond to exact fixed locations on the Earth.
For example, a floor plan map might specify that the southwest corner is 0,0 and the northeast corner is 500,500, so that if the building were moved to another location, these coordinates would still be accurate because they are not tied to exact locations on the Earth. If you used latitude/longitude coordinates for that map, however, the coordinates would change if you moved the building, because each pair corresponds to one exact point on the Earth.
Some types of arbitrary XY systems, such as the state plane systems, make use of both of these concepts. Each system represents a specific area, usually a state or a section of a state. Within that area, the coordinates are meaningful and represent exact locations on the Earth. So a point with the coordinates 100000,100000 always represents a specific location in the New York state plane system. However, if you change the system to another state plane system, all of the data moves to that area, where 100000,100000 represents a specific location in that area. In this way, the data is arbitrary XY, because the numbers themselves do not represent exact locations except within the context of the fixed areas.
Arbitrary XY data is not converted to latitude/longitude. Use it only in maps that use arbitrary XY as the system and in which all data is in arbitrary XY.
To set the coordinate system for your map