The five basic stands of interest to the Middle Ages are:
Marudai--round stand
Kakudai--square stand
Ayatakedai--bamboo design stand
Takadai--high stand
Karakumidai--Chinese or Foreign stand

On the Marudai the braid is made from the "bottom up to the top" and gathers under the stand. This handout deals with this style of stand but et up on a wooden or foam disk.  The other stands are shown for reference and 

On the Kakudai the braid is worked from the top to the bottom and is suspended over the top of the stand.  Braids made on this stand are hollow.

The Ayatakedai, Takadai, and Karakumidai can only do flat braids.  Each stand has different techniques and pattern mapping. 
I remembered to update my page on 5/26/2013
Beginning Kumihimo
The Stands
Above Left: My slotted disk that lasted for around five years
Above Right: My Tabletop Marudai 24 inches in diameter
Left: Kakudai
Right: Takadai fromt the back and from the front
Left: Wooden Ayatakedai
Right: My Ayatakedai made out of pvc pipe and paint stirring sticks
Left: Karakumdai