The Black Sheep is the one member of a group who does not fit in because they are seen as strange, unique, or undesirable in some way.
A Black Sheep is not just someone who doesn't fit in with their social circle or family — it's someone who rejects their role in the group, either deliberately or not. Or possibly someone whose reputation or morality is at odds with the family reputation/morality because it makes them different from the rest of the group.
Sometimes the Black Sheep is really bad — AKA a criminal. Other times a Black Sheep is just considered rebellious and might be a Cool Big Sis or Cool Uncle. On occasion, they are actually the Defector from Decadence. Often, they are a Misunderstood Loner with a Heart of Gold.
In kids shows, there's a common Aesop where a Black Sheep from one of the kids' families will show up as either a Cool Big Sis (or brother) or a Cool Uncle. Often the Cool Relative will inspire emulation by all the kids in town (sometimes excepting only the kid who is actually related to him). Eventually this emulation gets the kids into trouble. Either the Black Sheep helps the kids out and then delivers the Aesop that they really don't want to be like him, or he reveals himself to be a Jerkass and leaves the kids in their mess (often to be saved by the one kid who didn't emulate the Black Sheep).
Contrast White Sheep, where a family member is an outcast for not engaging in the family's poor behavior. An Oddball in the Series is often called its "black sheep". The Remittance Man is a specific example who mostly appears in The Western; an unwanted scion of a prominent European (usually British) family, sent overseas to make his fortune or to avoid scandal, and paid a regular sum of money (the "remittance") to stay there.
