A flat map is called a map projection. These maps are designed to take the spherical nature of the earth and portray it flat so it can be seen and analyzed for different purposes.
Creating a map projection has many uses, but also some disadvantages, including the fact that it will usually distort the realistic nature of the area it represents.
Map projections come in a variety of forms. Cylindrical projections have straight meridian lines and parallel lines. They are formed by taking the map and rolling it as if it were a cylinder. Pseudocylindrical projections are similar, but have straight longitudinal lines and curved meridian lines. Conic projections are taken by forming a cone of the area and distorting scale and distance, except along parallel lines.