Mr. Bear is Stephanie Tanner's teddy bear.
Stephanie got Mr. Bear when her mother came home from the hospital with baby Michelle. This is revealed in the episode "Goodbye Mr. Bear" where Michelle hides Mr. Bear and Danny decides to show the girls some old home movies. Jesse says, "Pam gave D.J. a charm bracelet, and she gave Stephanie Mr. Bear".
Mr. Bear suddenly disappears after Stephanie enters her teen years, and it was shown in several episodes that Michelle wanted him, so maybe Stephanie gave him to Michelle, or perhaps to Nicky and Alex. He was not mentioned until the season seven episode "The Apartment", which also marked his final appearance.