The Order of the Bear was an order of knights who believed that King Arthur would one day be reincarnated. The Order searched for Arthur's potential reincarnation every generation. Hoping to find the king before the reincarnation of Mordred appeared.
The "Order of the Bear" is a creation of Meg Cabot in the book "Avalon High". There are no references to any such order in any other Arthurian text, or any historical documents regarding Britain of the Dark Ages.