You died, or in the Horcrux case you get destroyed by the poison inside Gryffindor's sword. Harry has the antidote so he doesn't died, he's not destroyed beyond repair and the Horcrux stayed inside him. It's never stated that the substances to destroy the Horcruxes must be without any kind of possible antidote.
On destroying Horcruxes
Of course, the ever-wise Rowling was able to shed some light on the matter. “A Horcrux can only be destroyed if its container is damaged beyond repair, Harry was healed by Fawkes,” she wrote, referencing Dumbledore's beloved phoenix. “Had he died, the Horcrux would indeed have been destroyed.”
The diary was the first of Lord Voldemort's Horcruxes to be destroyed, by Harry Potter in the Chamber of Secrets, who stabbed it with the fang of the very Basilisk that was used to commit murder in the first place.
Well, mostly because he wanted the Chamber opened. He wanted to get someone into that place, he wanted to give them the diary that would take them into it.
However, the material or curse that is used to destroy a Horcrux must be something from which the Horcrux cannot heal. Basilisk venom is one of the few materials so potent that it can have this effect.
Voldemort intentionally made six Horcruxes, but when he used Avada Kedavra on Harry, he unintentionally created a seventh Horcrux. Instead of dying, Lily's love for Harry created a counter 'curse' known as Sacrificial Protection and saved Harry.
So why wasn't the Horcrux residing in Harry's head destroyed when he was bitten by the Basilisk? The answer is simple: He didn't actually die. Fawkes quickly healed the bite wound with his phoenix tears just in time. Case closed.
He finds out for sure immediately after Nagini's attack on Arthur Weasley. This night is significant in many ways. While Harry sleeps, he enters Nagini's mind.
Why did Dumbledore not tell Harry that he was a Horcrux? It appears that Dumbledore was afraid of Voldemort finding out that he knew about the Horcruxes. At the end of Deathly Hallow Harry realized that Dumbledore also did not want to waste anyone else's life.
Voldemort wasn't aware of Harry being a pseudo-horcrux because he didn't plan it, and Harry didn't know either until Voldemort “killed” him, but he actually killed the piece of soul kept in him.
Though Harry encountered many opportunities that deemed Avada Kedavra necessary, it remains as one of the Unforgivable Curses Harry Potter never cast. For one, he viewed the spell as an immoral practice commonly used by users of the Dark Arts.
So when Ron said three, it wasn't a mistake. He meant there were three more horcruxes to destroy before being able to destroy the fourth and final piece which was in fact Voldemort himself.
Quirrell is, in effect, turned into a temporary Horcrux by Voldemort. He is greatly depleted by the physical strain of fighting the far stronger, evil soul inside him.
Draco Malfoy's mother Narcissa was cold, cunning and devoted to the Dark Lord. But she was also a mother, which meant she was willing to risk everything to make sure her son was safe. When Harry survived Voldemort's Killing Curse for the second time, Narcissa pretended he was dead so she could get to Draco.
Dumbledore reveals how thoroughly he loves Harry. Harry is brave, and Dumbledore is proud as any father would be. More than that, Dumbledore succumbs to a parent's weakness; he cares more about Harry's happiness and wellbeing than the wizarding world he is trying to protect.
Question number one: Why didn't Dumbledore just give the sword of Gryffindor to Harry instead of leaving it to him in his will? He knew even before his death that Harry would need the sword to destroy Horcruxes, and he must have known that the Ministry would not let Harry get the sword.
She noticed how Harry acted more like Voldemort when the locker was on, while Ron had acted insecure and her emotional. She knew that something was off and at that point it was in her head that he was a horcrux.
“Dumbledore didn't want to lose his wand at that point and Draco disarmed him. So that meant that the wand gave Draco its allegiance, even though Draco never knew it, even though Draco never touched it. “From that moment on, that wand gave its allegiance to Draco, and it wouldn't work as well for anyone but Draco.”
James inherited the cloak from his father, Fleamont Potter, who inherited it as a descendant of Iolanthe Peverell -- a granddaughter of Ignotus Peverell. The cloak passed down through generations of the family until it reached James and then Harry.
Dumbledore knew Harry and Ron were under the invisibility cloak at Hagrid's hut in Harry Potter and the Chamber of Secrets, as his eyes flickered in their direction.
First of all, Dumbledore didn't reveal the secret of horcruxes to Snape because he didn't find it necessary. It wasn't his style to let out unnecessary secrets to people. Making horcruxes to fight horcruxes wasn't the plan.
Dumbledore had to keep certain things hidden from Harry just in case Voldemort read his mind. He did not want Harry knowing that Snape brought the sword of Gryffindor for that reason, it is very likely that he did not want Harry knowing that he planed to have Snape kill him for the very same reason. Save this answer.
To be on the safer side, he left the sword of Gryffindor for Harry on his will so that Harry, Ron, and Hermione can get a clue about a way to destroy the Horcruxes. Leaving Basilisk fangs on his will would have been a stupid decision as it would have given clue to the ministry about their plans.
Horcruxes can only be destroyed in specific circumstances that allow it's complete and utter destruction. In the movies, the Horcruxes are destroyed in these ways: The killing curse — “Avada Kedavra.” Fiendfyre — a curse that creates powerful and magical flames.
Gaunt's ring
After retrieving the ring Albus Dumbledore himself fell victim to a curse that, even when contained, would likely have killed him within the year. Once rendered defunct as a Horcrux, the stone was placed inside a Golden Snitch and left to Harry Potter in Dumbledore's will.