Simply put, the UVW modifier controls how your material gets to be applied on your object. There are many cases in which your texture will appear strecthed and that is becaose your 3D objects has no mapping coordinates, it doesn't know how to 'receive' a map (in a geometrical way). The UVW coordinates system is similar to XYZ coordinates, so essentially, the moddifier gives mapping coordinates to the object.
The mapping group defines the type of mapping coordinates used. It is based on how the map is projected onto the object: planar, cylindrical, cap, spherical, shrink map, box and face. Each will project the image in different ways on your object. Box might work on a sphere so give a try, experiment with it.
The length, width and height specify the dimensions of the UVW Map gizmo. A U, V, and W tile specify the dimensions of the map. You can tile the image and create a bigger, seamless texture if you need to without having to go back into Photoshop or an editing software to do this.