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I will assume that by the real world you mean the Earth.
I will also assume that you mean the whole Earth.

In this case a map is always a projection of a sphere on a plane (or a surface isomorphic to a plane) because a sheet of paper on which a map is drawn is necessarily a plane.
The mathematics you refer to are always simply a transformation of the natural spherical coordinates (e.g latitude, longitude) to the natural cartesian coordinates on a plane (e.g x, y).
That's why the first task is to decide what projection you will be using because it will dictate the transformation T such as T(lat,long) = (x,y).
Once you fixed T the process is straightforward.
You create a mesh of important points on the sphere (f.ex coastlines) with their coordinates (lat,long), then apply T on a point Ms (M on a sphere) , obtain the coordinates (x,y) of the corresponding point Mp (M on a plane) and draw it on the sheet of paper.
Then you just connect the points and obtain the coastline on a plane, e.g on a map.
Same method applies to rivers, mountains and any feature you want to represent.

As an example, the simplest projection is equirectangular (or cylindrical).
It is the simplest because your coordinates on the plane are also latitude and longitude.
That means that your sheet is divided from -180 to + 180 from the left to the right edge (e.g the horizontal axis covers 360° of the Earth's perimeter) and from -90 to + 90 from the bottom to the top (e.g it goes from the South pole to the North pole).
The map is therefore always a rectangle 2(horizontal) x 1 (vertical).
So for every point on the sphere of coordinates (lat,long) you simply draw a point of same coordinates (lat,long) on the map.
Of course this projection has an anomaly because to the 2 pole points is not associated a point on the map but a whole line (the upper and the lower edge of the map).

The corollary is that this map is easy to make but it deforms strongly shapes and areas when you approach the poles.
And it is only starting from this observation that some creative mathematics are necessary to find other projections that represent the shapes and areas better even near to the poles.
I will not develop farther because you asked only for a simple and general explanation.