Deedee is a brown squirrel and the 4th puppet who appears in the Puppet segments in season 1. She doesn't appear in all of them like Gilbert, Rexy and Teddy, but is still considered a main character by the producers of the show.
Deedee lives in the tree in Caillou's backyard and is often seen playing with Rexy, but sometimes interacts with Teddy and Gilbert. Deedee first appeared in a puppet segment when she was a baby squirrel lost from her family when Rexy found her. Deedee does have some family, which is she mentions she is having a family reunion with once.
Deedee did not appear in the third season puppet segments; she wasn't even mentioned, though the hole in her tree remained; this was never explained, though it is likely she moved. Another, darker theory a few older fans have is that she was eaten by a predator.