"So Death picked up a stone from the riverbank and gave it to the second brother, and told him that the stone would have the power to bring back the dead."
-- Beedle the Bard, Tale of the Three Brothers (TBB/TTB)
The second Peverell brother, described as “arrogant,” told Death he wanted to call people back from the grave, so Death gave him the Resurrection Stone (DH21, TBB/TTB). Cadmus used the stone to call back a young woman he loved and wanted to marry in life, but she was unhappy to be resurrected and remained “separated from him as by a veil.” He was so grief-stricken that he took his own life.
Cadmus was the ancestor of the Gaunt family through an unknown female ancestor, possibly a sister of Iolanthe Peverell. The Resurrection Stone was placed in the Peverell ring and eventually handed down to Marvolo Gaunt (HBP10, DH22). Marvolo thought the Peverell “coat of arms” was engraved on the stone, but it was actually the symbol of the Deathly Hallows.
- Marvolo's grandson, Tom Riddle, took his ancestor's Peverell ring and turned it into a Horcrux without realizing the stone was one of the Deathly Hallows. He was raised in a Muggle orphanage and never heard the Tale of the Three Brothers (DH22)
- Grindelwald hoped to use the Resurrection Stone to create "an army of Inferi," but he never found it because the Gaunts had it (DH35).
- Dumbledore called the actions of Cadmus "meddling in the shadowy art of necromancy" (TBB/TTB). Raising the dead is impossible even for wizards except for making Inferi which are only animated bodies. And according to Dumbledore, the young woman Cadmus called back was not real, but merely sent by Death to lure him to his own grave, which apparently worked. Yet having this correct opinion did not stop Dumbledore from trying to use the Resurrection Stone to see his sister and parents once again, and when he found the Peverell ring in the old Gaunt House, set with the Resureection Stone, he put it on too eagerly, forgetting that it had been turned into a Horcrux by Voldemort (DH35). Thus Dumbledore also marked himself for death, although Snape was able to postpone it for one year by sealing the curse in one hand (DH33).
- Dumbledore gave Harry the Resurrection Stone sealed in a Golden Snitch that only Harry could open using the flesh memory of having caught it during a Quidditch match (DH7).
Brothers: Antioch and Ignotus
Descendents: Corvinus Gaunt, Marvolo Gaunt, Morfin Gaunt, Merope Gaunt, Tom Marvolo Riddle (Voldemort)
Probably named for Cadmus, the founder and King of Thebes in Greek mythology, who killed a Water Dragon and sowed the teeth into a field to create an army of warriors. He sowed a jewel along with the teeth, and the majority of the warriors killed each other fighting for the valuable stone. Bad luck ensued from slaying the dragon and sowing its teeth, and Cadmus began to turn into a dragon or giant snake himself, along with his wife (Sources: Wikipedia a b). A number of similarities with the story of Cadmus Peverell are evident.
the majority of the warriors killed each other fighting for the valuable stone: Compare to the history of the Elder Wand and its owners - and to Chaucer's "The Pardoner's Tale," on which "The Tale of the Three Brothers" is almost certainly partly based. -BB
Cadmus began to turn into a dragon or giant snake himself, along with his wife: Perhaps a nod to the fact that the descendents of Cadmus Peverell - the Gaunts and Tom Riddle - were Parselmouths.