The Ring Bear is an episode from Season 2 that originally aired on September 11, 2000.
Miss Murray, Little Bill's teacher, announces to the class that she will be getting married to Dr. Clinkscales and would like two of the children to be attendants in her wedding. To be fair, she selects names from a hat, and picks Kiku to be the flower girl and Little Bill to be the ring bearer. Little Bill mishears the teacher and thinks his job will the "Ring Bear," meaning he will dress up in a bear costume. He's very disappointed when Brenda corrects him and tells him he'll have to wear a tuxedo, but she also makes him understand that he made a promise to Miss Murray to do his job at the wedding, even if it means he can't be a bear. He sadly goes upstairs in his room. In this episode, Bobby and Big Bill don't talk, and Alice the Great is not in this episode.
Mr. Rogers' Neighborhood had a similar issue like the one in this episode when Daniel Striped Tiger was invited to be a ring bearer in a wedding and also misheard and thought he had to wear a bear suit, just like Little Bill did.
On Urban Dictionary, Ring Bear is an actual bear or person dressed as a bear that presents rings at a wedding.