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.
We know for a fact that Basilisk venom is one of the few substances strong enough to destroy a Horcrux. Not only does it destroy the one inside Tom Riddle's diary, but Ron and Hermione also use one of the fangs to destroy Helga Hufflepuff's cup.
Along with the Cruciatus Curse and the Imperius Curse, Avada Kedavra is considered to be one of the most terrible curses in the magical world, called the Unforgivable Curses; the use of any of the three on another witch or wizard is punishable by a life sentence in Azkaban.
The Killing Curse rebounds on Voldemort during his duel with Harry by the green light of the curse emitting through the crack in the Elder Wand, causing Voldemort's hand to turn black and spread through his body, which in turn led to his death.
The Locket
Of all of Lord Voldemort's Horcruxes, Salazar Slytherin's old locket is the most difficult to obtain and destroy. In the Half Blood Prince, Harry Potter and Albus Dumbledore travel to an old cave where the Dark Lord would torture victims during his time as a teenager.
Hermione Granger and Ron Weasley destroy the Hufflepuff's cup Horcrux inside the Chamber of Secrets—Hermione giving it the fatal blow with a basilisk fang. In the Harry Potter and the Deathly Hallows book, Vincent Crabbe inadvertently destroys Rowena Ravenclaw's diadem in the Room of Requirement via Fiendfyre.
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.
Harry refused to sink to that level of violence because Voldemort was the epitome of evil, as was the Killing Curse, and he would not use the Unforgivable Curse that killed his parents. There was also a belief that the caster of Avada Kedavra needed the willingness to commit murder for the spell to work.
Cruciatus Curse ('Crucio')
Harry tried to use this spell himself. The first time was against Bellatrix after she had killed Sirius.
Harry makes use of two of the Unforgivable Curses in the books. Dueling with Bellatrix Lestrange, he attempts the Cruciatus curse, with limited results; Bellatrix says that he has to really hate someone to make the Unforgivable Curses work properly, righteous indignation isn't enough.
Expelliarmus did not beat Avada Kedavra.
Voldemort's wand (the elder wand) recognized its true master (that was Harry) and refused to attack it. In a way Voldemort killed himself as his own spell backfired on him.
In order to conjure the avada kedavra curse, you have to want to kill your victim. We all know that Voldemort could easily kill a child without an ounce of remorse... but not Snape. Snape didn't want to kill Dumbledore, and this was why the spell was blue instead of the usual green.
Although it might not have killed Nagini, the killing curse did succeed in destroying the Horcrux side of Harry.
Phoenix's tears heals wound. Objects doesn't have wounds. In order to destroy a Horcrux, its host must be destroyed beyond repair.
It is perhaps interesting that no person destroyed more than one Horcrux. Harry, Ron, Hermione, Neville, Dumbledore, Crabbe (indirectly: the diadem fell apart in Harry's hands after being damaged by Crabbe's fiendfyre), and Voldemort each destroyed one of the seven.
McGonagall followed up Harry's action by using the Imperius Curse on Amycus before tying him up with a net, rendering him useless during the Battle of Hogwarts.
The Killing Curse was created in the early Middle Ages by Dark witches and wizards. Primarily, the curse was used to quickly slay opponents during a duel. In 1707, the Wizards' Council was restructured into the Ministry of Magic, which allowed for more strict restrictions on certain types of magic.
Dumbledore, at one point, possessed all three Deathly Hallows. From Dumbledore's letters to Grindelwald in Deathly Hallows, it is obvious that the headmaster of Hogwarts was obsessed with the idea of the Hallows in his youth.
16. Avada Kedavra. J.K. Rowling has spoken about the root of this spell. According to Rowling, its root is actually Aramaic and derives from the original "abracadabra," which means "let the thing to be destroyed." In this case, the thing is a person.
Roughly translated, Expelliarmus – the Disarming Charm – means 'to drive out a weapon' and that's what it does: forces the subject to drop whatever they're holding. Usually that's a wand, which is why it's often seen in duels, and Harry definitely took that to heart, given how much he used it in combat.
First year
—Harry Potter's first lesson with Severus Snape. Students in their first year are taught the Boil-Cure Potion, called the "Cure for Boils" by Professor Horace Slughorn. This potion consists of dried Nettles, snake fangs, stewed horned slugs, porcupine quills and Flobberworm Mucus.
She was sorted into Slytherin House at Hogwarts School of Witchcraft and Wizardry and hated her time at the school due to never being given any positions of power. After her time at Hogwarts, Umbridge rose to prominent and influential positions in the Ministry of Magic in the Improper Use of Magic Office.
At the time Harry touched him, Quirrell was sharing a body with Voldemort. Since the magical blood protection Harry had prevented Voldemort from touching him, Quirrell would suffer pain. He would ultimately be destroyed from touching Harry.