in Half-Blood Prince, we find out that Voldemort's family ring was a Horcrux, and in Deathly Hallows we find out that it actually contained the Resurrection Stone. How did the Gaunt family end up with the stone? Are they direct descendants of the original Peverell brothers?
It's not clear how the Gaunts are direct descendants of the Peverells -- specifically, the Gaunts are proported to be descended from Cadmus Peverell, owner of the Resurrection Stone. Although, how a direct blood relation to Cadmus can be proven is tricky because, as The Tale of the Three Brothers in Tales of Beedle the Bard tells us, Cadmus used the stone to bring back from the dead the woman he was going to marry, but never did; he committed suicide almost immediately after bringing her back from the dead. The assumption would be that they did not have children -- so, who then did the ring/stone pass on to? We know Antioch Peverell was almost immediately killed by a rival seeking the Elder Wand. So that leaves Ignotus Peverell, owner of the Invisibility Cloak, as rightful heir to the stone. But Beedle the Bard never says that Ignotus claimed the stone or kept it to pass down to his own son.
It's possible that Ignotus claimed the stone after Cadmus's death and that he did indeed pass it down to his son. According to J.K. Rowling, Harry is a descendant of Ignotus Peverell (so is rightful heir to the Invisibility Cloak) and is distantly related to Voldemort through the Peverell line. I haven't read anything more specific than that, though. So will we know how the Gaunt family came into possession of the stone/ring? No. At least not until J.K. Rowling tells us (Anyone, please correct me if I'm wrong and she's said something about it.). If there is a distant relationship between Harry and Voldemort's families, the possibility exists that at some point centuries ago, Gaunt's side of the family somehow got the stone from the Potters' side of the family, but to say how would be pure conjecture.
Also, if you read the text carefully, it has Morphin Gaunt claiming to be the last remaining descendant of Salazar Slytherin; he never claims to be related to the Peverells, only that he is in possession of the stone/ring with the Peverell coat of arms engraved on it (aka the sign of the Deathly Hallows) and that it has been in his family for centuries.
For all we know, at some point a Gaunt stole the stone and began passing it off as a family heirloom to bolster their pure-blood pride. But, again, this is pure conjecture and there's no canon veracity to this idea. It's just not clear, unfortunately.