In To Kill a Mockingbird, Uncle Jimmy is James "Jimmy" Hancock, the husband of Atticus's sister (and Scout's aunt) Alexandra Finch Hancock. In the course of the novel, Scout mentions that she has never taken notice of Uncle Jimmy because the only words he ever spoke to her were those telling her to get off a fence. Scout mentions that Alexandra herself did not take much notice of Uncle Jimmy either, suggesting that they do not have a happy or close relationship in their marriage. Uncle Jimmy is the father of Alexandra's son, Henry, and the grandfather of Henry's son, Francis. When Aunt Alexandra goes to Maycomb, Uncle Jimmy does not go to Maycomb with her but, instead, stays at home. When Scout asks Alexandra whether she misses her husband, Alexandra doesn't respond. This leads Scout to state that "Uncle Jimmy present or Uncle Jimmy absent made not much difference, he never said anything." For Scout, Uncle Jimmy is a nondescript, silent man who seems to be of virtually no importance to her, his wife, or the story itself.
You're right, some of these minor characters in To Kill a Mockingbird are hard to keep straight. As a precursor, I should mention something about the Finch family. Put simply, Atticus has one brother and one sister: Jack and Alexandra. These are Scout's "Uncle Jack" and "Aunt Alexandra."
Uncle Jack is NOT Uncle Jimmy!
Aunty [Alexandra] and Uncle Jimmy produced a son named Henry, who left home as soon as was humanly possible, married, and produced Francis. Henry and his wife deposited Francis at his grandparents' every Christmas, then pursued their own pleasures.
Let's make it simple. Uncle Jimmy is Aunt Alexandra's husband. In fact, Aunt Alexandra's married name is Alexandra Finch Hancock (because she married Jimmy Hancock). Therefore, he is not a "blood" Finch, but married into the family. Alexandra and Jimmy live at "Finch's Landing," which is the family plot of land. Aunt Alexandra and Uncle Jimmy have one son: Henry. Henry is grown up, gone form home, married, with a son of his own: Francis.
Uncle Jimmy is the anti-social husband of Aunt Alexandra. He lives in the Finch family house with Aunt Alexandra out at Finch's Landing. Scout doesn’t know too much about him because she never hears him talk. Once, though, he did scold Scout for climbing a fence, but, other than that, he stays to himself. Aunt Alexandra doesn’t even talk about him and seems somewhat distant in her feelings for him and cool toward their relationship. When Aunt Alexandra comes to Maycomb to watch Scout and Jem, Uncle Jimmy stays home and doesn’t accompany her. Uncle Jimmy is just another “odd” eccentric character who shows the variety of Maycomb’s citizens. Like Miss Tutti and Miss Frutti, a pair of eccentric sisters, Uncle Jimmy is another colorful character who adds a little more atmosphere to Lee’s story.