Abbie is the daughter of Edwina, a woman with whom Abraham Simpson briefly had a relationship while he was stationed in England during World War Two. It is insinuated that Abbie is the result of that relationship, which would make her Abe's daughter and Homer's half-sister. She looks and acts very similar to Homer.
Abbie was the result of a brief relationship between Edwina and Abe during the Second World War. Abe was in England before D-Day and had had one night of passion with Edwina before he had to leave England. Unknown to Abe, Edwina became pregnant and the result was Abbie.
Abe and the rest of the Simpson family first met Abbie when she was in her 50s. They had been visiting England and were on their way to catch their plane back to America, and Edwina and Abbie approached them in the airport. When Abe realized that Abbie was his daughter, he ran off to the plane screaming, after saying "See you in Heaven."
While Abbie is obviously a biological Simpson (aside from her appearance and personality, she was born less than a year after Grampa's affair with her mother) she is not officially recognized by the family and never explicitly called a Simpson.
It is possible that even though she is female, Abbie may have somehow inherited the Simpson Gene defect that most of the Simpson males have. As Dr. Simpson pointed out, all the Simpson women have high intellects and good careers, because the defective gene is on the Y chromosome. This could be an instance of the gene skipping a generation or striking some and missing others, as her half-brother Herbert Powell also supposedly inherited the Simpson Gene, but it has not significantly affected him.