Who are Alexandra, Jack and Francis, and what is their relationship to Jem and Scout in To Kill a Mockingbird?
This above answer offers a good basis. However, a few more points could be made.
Yes, Alexandra is Scout and Jem's aunt, but she is also described as very proper, traditional, and well-mannered. She tries to impose this decorum upon Scout. At times she disapproved of how Atticus was raising his children. Here is what Scout says of her:
Aunt Alexandra was one of the last of her kind: she had river-boat, boarding-school manners; let any moral come along and she would uphold it; she was born in the objective case; she was an incurable gossip.
Uncle Jack was beloved by all. He is significant, because he bought air rifles for Jem and Scout for Christmas. Needless to say the children loved these guns. But it was in this context where we learn about the title of the book, because Atticus says that it is a sin to kill a mockingbird. So Scout and Jem learn a life lesson.
As for Francis, he was the son of Jack and Alexandra. Scout did not have many nice things to say about him. He and she did not mix at all. At one point she even thought about shooting him with her new air rifle.
ALEXANDRA FINCH HANCOCK. Alexandra is Atticus's older sister, making her the aunt of Jem and Scout. She is also the sister of Jack Finch and grandmother of Francis Hancock. She is married to Jimmy Hancock (the uncle of Jem and Scout); their only son is Henry, father of Francis.
JACK FINCH. Atticus's younger brother, Jack is a bachelor physician in Boston. Jem and Scout always look forward to seeing their Uncle Jack and hearing his stories at Christmas.
FRANCIS HANCOCK. The obnoxious Francis is the son of Alexandra and Jimmy's only son, Henry Hancock. "The most boring child I ever met," Francis is a favorite of Alexandra and Scout's sworn enemy. Considering his name, tiny stature, love of fancy clothing, interest in cooking, and the fact that Scout could "split my knuckle to the bone on his front teeth," it is clear that Francis is a character whose future includes gender issues.