James is the protagonist of The Snowman (which is based on the book of the same name by Raymond Briggs) and a minor character in Father Christmas. In the 1982 film, his "Older James" voice --heard in the introduction (and the only dialogue in the film)-- is voiced by Raymond Briggs (who is also the author of the book version).
In The Snowman, James is a young boy who lives in the English countryside with his parents. One snowy Christmas Eve, he builds a snowman which comes to life and takes him to the North Pole where they attend a snowman party and meet Father Christmas and his reindeer. James is given a snowman scarf by Father Christmas before returning home with his snowman. When he and the snowman get back to James's house, the duo do a farewell hug before James goes back into the house and to bed. The next morning, James wakes up. He gets dressed, runs down the stairs, and goes past his parents (who are seen eating breakfast). When he gets to the garden, he finds the snowman. That is, only to find that the snowman is melted. James gets sad, takes out his scarf, and mourns the loss of the snowman (since the snowman is gone). That is, before the end credits roll.
The snowman is melted. But James, he builds his snowman again the next year's winter. And on Christmas season (Christmas Eve) he and his snowman go to the North Pole. This happens in the sequel Father Christmas.
James reappears in Father Christmas (which takes place roughly a year or so after the events of The Snowman) where he and his snowman attend another snowman party at the North Pole and meet Father Christmas again. They then find two presents in Father Christmas's sleigh that haven't been delivered. Father Christmas leaves the party to deliver the presents, which are revealed to be for the British Royal Family.
James makes a cameo appearance in The Snowman and the Snowdog where he appears in a photograph found by Billy (the protagonist of the film, who now lives in James's old house).