Mapping Coordinates

Decoration on the vase is a map positioned by rotating the Texture Map Modifier gizmo.

Mapping coordinates specify the placement, orientation, and scale of a map on the geometry.

If you apply a mapped material to an object that has no mapping coordinates, the Renderer assigns default mapping coordinates. The built-in mapping coordinates are designed for each object type. The box mapping coordinates place a duplicate map on each of its six sides. For the cylinder, the image is wrapped once around its sides, and duplicates of the image are distorted at the end caps. A sphere has the image wrapped once around the sphere, and then gathered at the top and bottom. Shrink-wrap mapping is also spherical, but truncates the corners of the map and joins them all at a single pole, creating only one singularity.

There are three ways to apply mapping coordinates:

Set the Generate Mapping Coords option in the creation parameters rollout of any standard primitive. This option provides mapping coordinates specifically designed for the primitive. Because mapping coordinates require additional memory, turn them on only when you need them.
Use special mapping coordinate controls for special objects. For example, Loft objects provide built-in mapping options that let you apply mapping coordinates along their length and around their perimeter.
Apply a Surface Mapper modifier. This world-space modifier takes a map assigned to a NURBS surface and projects it onto the modified object or objects. Surface Mapper is especially useful for seamlessly applying a single map to a group of surface sub-objects within the same NURBS model. You can also use it for other kinds of geometry.

There are three cases where you don't need mapping coordinates:

Reflection/refraction maps and environment maps
These use an environmental mapping system, in which the placement of the map is based on the rendered view, and fixed to the world coordinates in the scene.
3D procedural maps (such as Noise or Marble)
These are procedurally generated, based on the local axis of the object.
Face-mapped materials
The maps are placed based on the facets in the geometry.