Plaiting is most similar to a checkerboard weave.  A weft is woven over two warps at right angles to create a criss-cross pattern.  The result looks like a checkerboard.  This is most popular when using a cane or bark weave.  The larger strips allow for more coverage with less material.

With twining, a very rigid warp of twigs or roots, has a diagonal weft that is wrapped or 'intertwined' to create a basket that is very strong.  The emphasis on the strong foundation with a more flexible weft creates a very strong container.

Wicker is created with 'twig' like foundation more so than a grass foundation.  The thicker foundation (warp) is interwoven with a weft of twig like material creating a vessel that, though not water tight, is extremely strong.  The materials used are most often roots, vines, or yucca shoots.  Very popular with the Hopis.

 Materials in American Indian Baskets
 
The basket materials of the basket are typical of the plants surrounding the Native Americans.  this may be your first hint.  Is it a grass, a root, or a bark?  What material is used as a weft and what material is used as a warp? 

Designs and Patterns in American Indian Baskets 
Additionally, there are design cues.  The Navajo wedding basket is probably the basket that is most associated with Native American basketry. The Hopi baskets, with their colorful pictorials and/or kachina masks are likewise recognizable.  California tribes' basketry is also highly sought after.  The subtle patterns woven tightly and evenly.  And of course, the Apache pictographs.
Though Eastern, Plains, and Pacific Northwest tribes all created baskets, this website focuses more so on the West and Southwest tribes. 
 

Today, Navajo baskets from the Black family is an example of a highly stylized art form.  Today's basket weavers, though few in number, are some of the most technical