There are two ways in which you can manipulate mapping.
In this example a texture is placed into the Square Texture space (left), an UV Map of the object lays on top (Center) and a correspondence is set between faces of the object and colors of the texture (right).
01 – The first method is to move, rotate and scale the Map, so to vary the correspondence between Texture space and Map.
02 – The second method is to move, rotate and scale the Texture: so to vary the correspondence between Texture image and Texture space.
Pressing the Mapping button lets you use the method number 01: manipulating the corrispondence between Texture space and Map.