A member of a middle class or upper class family whom unlike other family members, could not get it together, and ended up a drug addict, a chronic alcoholic or incarcerated. It can also be used to describe a family member who is constantly unemployed and borrowing money with no intentions of returning it. There seems to be one of these in every family, no matter how perfect or well off the family makes themselves out to be. Note, that this definition does not cover family members who suffered addictions and did the responsible thing to overcome them, nor does it cover those who were wrongly imprisoned for something they didn't do or those who were laid off and constantly looking for work.
The family fuckup cannot be reasoned with. Their mindset is that the world is out to get them and that their lack of ambitions is somehow the fault of society. As a result, confronting the family fuckup results in them displaying aggressive, antisocial or psychotic behaviour 99% of the time.
Bob: man, I just gave my brother Joe $300 so he won't get evicted and I know he's not gonna pay me back.
Tom: yup, and he's probably gonna spend it all on booze.
Bob: yea the rest of us kids went to college and made something of ourselves, but not Joe. He's the family fuckup.