The Wolf was an elf and one of Santa Claus' helpers. He and his colleague Ian followed Santa in his sleigh on the way to deliver presents to children all over the world during Christmas.
He mocked Clara Oswald's belief that it was her mum and dad who once a year gave her a great, big pile of presents, and insisted that Santa was real.
He followed Santa to the North Pole to help out during the dream state of the Twelfth Doctor, Clara and four others being attacked by dream crabs, and later when Santa revealed that they were all dreaming. (TV: Last Christmas)
A running joke with him was his insistence on being politically correct.