A map projection is the result of applying mathematical formulas relating spherical coordinates in a system of geographic coordinates, to plane coordinates. These formulas transform a representation of the earth on a rotating ellipsoid into a flat representation, a map.
Below, the LAMBERT conformal conic projection. A cone is placed tangentially to the sphere. The 3D points on the sphere are projected on to the cone, then the cone is opened out giving a map with 2D points (and altitude information).