The Deathly Hallows are the three magical objects that are the focus of Harry Potter and the Deathly Hallows. The form, function, and purpose of each of the Hallows are revealed as the plot of the novel progresses. In the course of their investigation into Voldemort's Horcruxes, Harry, Ron, and Hermione speak with Xenophilius Lovegood who explains that the search for the legendary Hallows is referred to as the Quest. Also, very few actually believe this story, though many, like Viktor Krum, believe the sign of the Deathly Hallows to be the mark of Gellert Grindelwald.
According to The Tale of the Three Brothers, the Peverell brothers found Death. Death gave them a choice of anything they wanted; the first brother chose a wand that could not be defeated in battle, the second asked for a way to bring back someone from death, and the third selected a cloak that made the wearer invisible to hide from Death himself. According to Rowling, the story about how these objects came into existence is based upon Geoffrey Chaucer's The Pardoner's Tale.
Elder Wand

The power of the Elder Wand was first shown in the story, as Antioch Peverell, the first and oldest of the mythical Three Brothers, had a duel with an enemy he had long wanted to defeat. He won, and left his enemy dead on the floor.
After boasting of his unbeatable wand, Peverell was murdered in his sleep by a rival wanting to claim the wand. Ever since, power-hungry wizards have sought the wand. It eventually came to the possession of Gregorovitch, a Bulgarian wandmaker. Gregorovitch boasted about possessing the Elder Wand, believing it would boost his popularity, and he tried to reverse engineer its secrets as he faced competition from Ollivander. It subsequently fell to Gellert Grindelwald, who stole it from Gregorovitch. Ultimately Grindelwald was defeated by Dumbledore, who then assumed control of the wand, it being the "only hallow [he] was fit to possess, not to boast of it or kill with it, but to tame it."
When Dumbledore arranged his own death with Severus Snape, he meant for Snape to "end up with the Elder Wand". Because his death would not have been the result of his defeat, Dumbledore hoped this might break the wand's power. However, since Draco Malfoy had disarmed Dumbledore, the plan failed and Draco unwittingly became the wand's new master. After Dumbledore's death, the wand was placed inside his tomb.
In the final book, Voldemort learns about the wand and goes on a search for it, and eventually knows that Dumbledore possessed the wand. He opens Dumbledore's tomb and claims the wand as his own. Only later does he learn that he cannot be the master of the wand, as he did not gain the ownership from its previous owner. Assuming Snape is its current master, Voldemort slays Snape, not realizing that the wand's allegiance was to Draco (even though Draco never had the Elder Wand physically in his possession). Furthermore, Harry later disarmed Draco and took his wand (although that was not the Elder Wand), and thus the Elder Wand's allegiance shifted to Harry, before Voldemort took physical possession of the wand.
In the Battle of Hogwarts, Voldemort casts his final Killing Curse against Harry's Expelliarmus charm. But since the Elder wand's allegiance is to Harry, Voldemort's spell backfires and kills him.
The Elder wand was also used by Harry to repair his damaged holly and phoenix feather wand. Afterwhich, he returns it to Dumbledore's tomb in the hope of fulfilling Dumbledore's original plan: for the reigning owner of the Elder Wand to die a natural death, thus ending its bloody trail of violence.
Rowling revealed in an interview that the first working title for Harry Potter and the Deathly Hallows was Harry Potter and the Elder Wand.
Resurrection Stone

Dumbledore recovered the ring from Marvolo's estate, recognizing it as both a Horcrux and one of the Deathly Hallows. Forgetting that it was a Horcrux, and that Voldemort likely cursed it when he turned it into one, and motivated by personal desire, Dumbledore attempted to use the Resurrection Stone to talk to his deceased family. However, the curse destroyed his hand and began to spread throughout his body. Though the spreading was partly contained in the destroyed and blackened hand by Snape, Dumbledore was doomed, having at most a year to live.
The stone was later passed to Harry through Dumbledore's will, hidden inside the Snitch that Harry caught with his mouth, nearly swallowing it, in his first-ever Quidditch match. The Snitch revealed the message "I open at the close" when touched by Harry's lips. Harry is unable to open the Snitch until he is about to die, and he realizes that "the close" means the end, or his death. Harry uses the stone to summon his parents, Sirius, and Remus Lupin to comfort him before he meets Voldemort. The stone slips through Harry's numb fingers in the Forbidden Forest. He and Dumbledore's portrait later agreed that Harry would neither search for it nor tell others where it is. In a 2007 interview, Rowling said she would like to believe that a centaur's hoof pushed it into the ground, burying it forever.
Cloak of Invisibility

While making the wearer invisible to Muggles and wizards, some creatures are able to sense people hidden under it. Snakes, for example, cannot see through the Cloak of Invisibility, but they can sense movement and heat, and therefore can detect people under it. Mrs. Norris, Filch's cat, also seems to see Harry when he wears the cloak. Wearers can also be detected by the "Homenum Revelio" spell. In Goblet of Fire, Moody's magical eye could sense Harry under the cloak. In the Prisoner of Azkaban, Dumbledore warns that the Dementors' perception of humans is unhindered by invisibility cloaks, as they sense people through emotions.