There isn't too much known about Sherlock's family, but here's what I've found, based on canonical evidence and my own research.
The earliest known ancestor of Sherlock Holmes is the French painter Claude-Joseph Vernet. Vernet was Sherlock's great-grandfather. He had at least two children, Carle Vernet (who was also a painter) and an unnamed daughter (who was Sherlock's grandmother).
It is not made clear whether this French grandmother was on Sherlock's mother's side or his father's. The names of Sherlock's parents are unknown. Canon dictates that he had one brother, Mycroft Holmes, who was seven years his senior.
Only one other Holmes relative is mentioned in the canon. After Sherlock returns from "The Great Hiatus" (the three years after his supposed death on the Reichenbach Fall), he persuades Watson to sell his practice and move back to Baker Street. Watson eventually learns that the man who bought the practice, Verner, was a distant relation of Sherlock's. How exactly they were related is not stated, but the similarity in surname suggests that this Verner was probably some sort of cousin through the Vernet line.
Canonically, Sherlock never married and had no children. However, there is one interesting point made in another fandom which suggests that Watson didn't tell us everything about his friend. In the fourth Star Trek movie, Mr. Spock says, "According to an ancestor of mine, once you have eliminated the impossible, whatever remains, however improbable, must be the truth." This is one of Sherlock's favorite maxims, the implication therefore being that Spock is somehow descended from Sherlock. (Spock's mother, after all, was a human!) However, as has been suggested to me, the implication might be that Spock is descended not from Sherlock, but from his creator, Sir Arthur Conan Doyle.
(Right: Sidney Paget's drawing of Mycroft Holmes, from "The Greek Interpreter," as published in the Strand magazine, 1891)