"Centuries [this ring's] been in our family, that's how far back we go, and pure-blood all the way! We're [Salazar Slytherin's] last living descendants, what do you say to that, eh?"
-- Marvolo Gaunt (HBP10)
The Gaunts are an ancient, in-bred, Pure-blood Wizarding family, direct descendants of Salazar Slytherin. The family is listed in the infamous Pure-Blood Directory from the 1930s as being one of the “Sacred Twenty-Eight” families which stayed true to pure-blood status.
Most of the Gaunts were gifted with the ability to speak or at least understand the snake-language Parseltongue, making the family line unique in the Wizarding World. Because they were Parselmouths, the Gaunt heirs of Slytherin were entrusted over the centuries with the secret of how to open the Chamber of Secrets under Hogwarts Castle. A student named Corvinus Gaunt incorporated the trapdoor entrance into the plumbing of a new bathroom that was built in the 18th century, which orphan Tom Riddle re-discovered in the 20th century (Pm:Chamber of Secrets).
Family members in England during the 20th century included father Marvolo Gaunt, a son Morfin, and a daughter named Merope who lived living in slovenly conditions in a dilapidated shack on a wooded hillside outside of the village of Little Hangleton. Once a wealthy family, by the 1940s the Gaunts were best known for their instability and violent behavior. Most of what we know about them comes from Dumbledore’s interview of Ministry of Magic investigator Bob Ogden (HBP10). Ogden went to the Gaunt house in September of 1925 to serve a summons on Morfin Gaunt for attacking a local Muggle from a nearby manor house, Tom Riddle.When Harry saw Ogden’s memory in the Pensieve, there was a snake nailed to the door. Inside there are three very small rooms. The main room was filthy and squalid, and inside was Merope Gaunt, a daughter who had never attended Hogwarts.
Although treated as a non-magical Squib by her father, Merope Gaunt was actually a crafty witch. When she fell in love with Tom Riddle, she used a love potion to trick him into marriage and ran away from Little Hangleton with him. She soon became pregnant, and at some point, according to Dumbledore, stopped giving her husband the potion, which caused Tom Riddle to abandon his wife and child, and flee back home to his parents. Heartbroken and alone except for her unborn child, Merope sold Salazar Slytherin’s locket at Borgin and Burke’s, and gave birth to her son. Tom Marvolo Riddle, at a London orphanage before dying on New Year’s Day (HBP13). Tom Riddle grew up in the orphanage not knowing much about his ancestry, although he realized early on that he could speak to snakes (HBP13). After entering Hogwarts, he discovered the entrance to the Chamber of Secrets, releasing the Basilisk that killed student Myrtle Warren, who became the ghost “Moaning Myrtle” (CS17). Many deaths followed, including the murders of his own father and grandparents in Little Hangleton, and over many years Riddle transformed himself into Lord Voldemort, one of the most evil wizards of all time.
Although Lord Voldemort was vanquished twice by Harry Potter, once in 1981, and again at the Battle of Hogwarts in 1998, there were always rumors that he had left behind another Heir of Slytherin/Gaunt. While Draco Malfoy’s son Scorpius was the focus of most gossip, the real child of Lord Voldemort was hidden away at the home of Death Eater Euphemia Rowle (CC3.16). The girl – Delphi – whose mother was Draco’s aunt, Bellatrix Lestrange, attempted to use an Experimental Time-Turner to return to the past and somehow make sure the Dark Lord survived into the 21st century. But her plan failed due to intervention by Draco, Harry Potter, and others, and she was captured and sent to Azkaban (CC4.13).
The Gaunt Family in Ireland and America
Another Gaunt descendant was the Irish witch Gormlaith Gaunt, who killed her Muggle-loving sister Rionach Sayre and brother-in-law William Sayre before kidnapping her niece, Isolt Sayre, holding the girl hostage in Hag’s Glen. Eventually, Isolt rebelled against her cruel aunt, and after stealing Gormlaith’s wand, which once belonged to Salazar Slytherin, she sailed across the Atlantic in 1620 on the Mayflower to Massachusetts, where she and her Muggle husband founded the magical Ilvermorny School, named for the farm where her parents died (Pm: Ilvermorny). Gormlaith later discovered Isolt’s whereabouts, and infuriated that her niece married a Muggle, she came to America and attacked Ilvermorny, putting the family and Slytherin’s wand into Enchanted Sleep using Parseltongue. But she was defeated several different ways: the Horned Serpent cores inside the wands of Isolt’s adopted sons heard her speaking Parseltongue and sent out a warning sound; the cries of Isolt’s twin daughters Martha and Rionach awakened the parents because of the magic of Love; and finally, a friendly Pukwudgie named William jumped in the window and killed Gormlaith with a poison arrow (Pm). Slytherin’s wand was buried on the grounds of Ilvermorny and soon grew into a large snakewood tree.
To be gaunt is to be "very thin especially from disease or hunger or cold." (Wordnet)
The book Dominating Dementors: A True History of Azkaban which was found in Hermione's bookshelf in the Minister for Magic's Office, has a poem that mentions the Gaunt Family and seems to apply to both Tom Riddle and his daughter Delphi:
I was born in a cage
But smashed it with rage
The Gaunt inside me
Riddled me free
Of that which would stop me to be (CC1.19).