Master Micawber is Mr. Micawber's oldest son. He's a restless, smart kid who doesn't really appear much in the book. Mr. Micawber wants him to sing in choirs, but what he actually seems to enjoy doing the most is singing in pubs: "in short, he has contracted a habit of singing in public-houses, rather than in sacred edifices" (52.185). Obviously, Mr. Micawber's wandering here and there trying to avoid collection agents hasn't given his kid a strong taste for the church life. But it seems that Master Micawber shapes up in Australia.
(By the way, we have to note: "Master" is a common form of address for someone else's son in the nineteenth century, like Mr. or Miss or whatever we use these days. So Master Micawber isn't master of anything really; this just means that he's Mr. Micawber's son.)