The Crisis on Infinite Earths changed DC Universe continuity in many ways.
In the new continuity, Barbara was born to Roger and Thelma Gordon, and is Jim Gordon's niece.
(In Batman: Gotham Knights #6, Batman discovered a letter that Barbara had been keeping in a safe deposit box that had been stolen. The letter revealed that Barbara knew Jim Gordon had dated her mother Thelma before she married Roger, and that there is a chance that Jim Gordon might be her biological father, although he is not aware of that. Barbara has stated that she has not confronted Jim about this, not because she is afraid it might be true, but because she is afraid it might not be).
She showed interest at a young age in superheroes, often dressing up as one. Barbara's mother and aunt were killed in a car accident, partly caused by her father's drinking. Roger began drinking heavily and started taking his aggression out on Barbara. Barbara took on a great deal of responsibilities to cover for her father, consoling herself by thinking her father was not the same man she had once known. Eventually Roger's drinking and self-abuse caused his death.
James Gordon adopted his orphaned niece. James "Jim" Gordon, his wife (also named Barbara), and their son lived in Gotham City, where he worked as the commissioner of police. Although James and young Barbara initially did not get along, with time they grew close (Secret Origins #20) - Barbara came to think of him as her father (which he already was, legally).
http://en.wikipedia.org/wiki/Barbara_Gordon#Crisis_On_Infinite_Earths