We don't know a whole lot about this family, but this is what we've been able to pick up on so far.
Lily Evans Potter
Lily was a muggle-born witch who grew up in a small family with two proud parents and one jealous sister. As a small girl, Lily learned she was a witch from none other then Severus Snape who lived nearby. Lily's parents were thrilled when they learned of their daughter's magical abilities. She never failed to disappoint them either, as she was top of her class at Hogwarts. Lily, like her son, was sorted into Gryffindor. It was at Hogwarts that she met her future husband, James Potter. Lily had red hair and green eyes, exactly like Harry's eyes.
The Potters lived in a small cottage in Godric's Hollow. When Dumbledore learned that Voldemort planned to attack them, they were put under the protection of a Fidelius Charm. This magic failed when their secret-keeper, Peter Pettigrew, betrayed them. Although Voldemort would have allowed Lily to live (probably because Snape had asked for her to live), she refused to step aside and died protecting her son. Her death invoked an ancient magic that Voldemort did not comprehend, and it protected Harry from death.
It is James that Harry most looks like, though he has his mother's eyes. James was a popular student at Hogwarts who formed a tight knit group with Sirius Black, Remus Lupin and Peter Pettigrew. The four were always seen together, sometimes tormenting fellow students (such as Severus Snape). It was this behavior that disgusted Lily. It took a couple of years at Hogwarts before James was able to show that he could be a serious student. The two began dating in their last year at Hogwarts while Lily was Head Girl and James Head Boy.
Besides looks, James also shares some other similarities with his son. Both were in Gryffindor and played on the Gryffindor Quidditch teams. James (along with Lily) defied Voldemort on three occasions (unfortunately we don't know much about those incidents).
James was unarmed the night Voldemort attacked, and was killed just after he told Lily to take Harry and run away.
Ginny, the youngest of the Weasleys, married Harry at some point after Voldemort's defeat. To read more about her, visit the relationships page.
Harry and Ginny have three children named James, Albus Severus and Lily.
Sirius may not be blood kin, but it hardly seems to fair to put him anywhere else but here. Sirius came from a notorious pure-blood family, the Blacks. Sirius disagreed with his family on their attitudes towards muggles and half-bloods, and became an exile from the family. James Potter's family came to be more of a family to him then any other. He was James' best man at the Potter's wedding. As such, it was no surprise that James and Lily made Sirius Harry's godfather. After the Potter's were killed, Sirius was never given a chance to fulfill his duties as a godfather. He was framed for the murder of a dozen muggles by Peter Pettigrew. This forced him into a life sentence into Azkaban. During Harry's third year, Black escaped once he learned that Peter was at Hogwarts (in the form of Ron's rat, Scabbers). When the truth came out, Sirius took his rightful place as Harry's godfather. He gave him a Firebolt for Christmas to make up for being an absent godfather. Harry continually turned to Sirius for advice, up until Sirius' death at the end of Harry's fifth year.
Petunia Evans married Vernon Dursley some time before Lily's own wedding to James Potter. Two couples couldn't be more unalike, except for the fact that both had a single son. After the death of the Potters, Dumbledore arranged for the Dursleys to take care of Harry. They did so with as much spite and bitterness as they could muster. As such, Harry lived a terrible life with them. Going to Hogwarts provided the first real joy to him as it presented to him a life so different then he had before. Harry was, however, forced to stay with the Dursleys every summer at Dumbledore's request. This insured that the protection from his mother's blood would continue to last until his seventeenth birthday. The summer before he turned 17 was Harry's last known time spent with the Dursleys. Due to their relationship with Harry, the Order decided to hide them. Dudley and Harry spoke one last time, with Dudley telling Harry that he didn't think Harry was a waste of space after all.
Not much is known about the rest of Harry's family. We know he saw them in the Mirror of Erised, but most of his attention was turned to his parents. We do know that Lily's parents were delighted that she was a witch, and that James' parents were loving folks (who were especially kind to Sirius).