"To Ronald Bilius Weasley, I leave my Deluminator, in the hope that he will remember me when he uses it."
"To Miss Hermione Jean Granger, I leave my copy of The Tales of Beedle the Bard, in the hope that she will find it entertaining and instructive.”
“To Harry James Potter ...I leave the Snitch he caught in his first Quidditch match at Hogwarts, as a reminder of the rewards of perseverance and skill."
-- from The Will of Albus Dumbledore (DH7)
On Harry’s seventeenth birthday, Minister for Magic Rufus Scrimgeour showed up at the Burrow with Dumbledore’s last will and testament in which he bequeathed items to Harry, Ron and Hermione (DH7).
Most of Dumbledore's possessions, including his library and the silver instruments in his office, were left to Hogwarts (DH7).
Scrimgeour was suspicious of Dumbledore's plans, and had held back the will for 30 days while the Ministry studied the objects mentioned. He justified the delay by citing The Decree for Justifiable Confiscation, but Hermione pointed out the law was "created to stop wizards passing on Dark artifacts" (DH7). Scrimgeour had to give up once the thirty days were over since he had no proof the objects were dark.
To Ron, Dumbledore left his Deluminator, the instrument he once used on Privet Drive to put out the Muggle lights while waiting for Hagrid to arrive with Baby Harry (PS1). Ron later used it to help him in his search for Hermione and Harry in the Forest of Dean, as it created a ball of light that entered his heart and helped him know what to do and where to go (DH19).
To Hermione, Dumbledore left his ancient copy of Tales of Beedle the Bard with a worn and stained cover. The title was written in Runes, which Hermione had studied at Hogwarts. As Hermione began to cry over the bequest, Scrimgeour wondered if she and Dumbledore were using secret codes in the book, to which she replied sarcastically: "if the Ministry still hasn’t found any hidden codes in this book in thirty-one days, I doubt that I will" (DH7). Hermione didn't believe in the existence of the Deathly Hallows until they talked to Xeno Lovegood and Harry convinced her he posssessed the real Cloak of Invisibility from the Tale of the Three Brothers, and possibly the Resurrection Stone inside the Snitch (see below) (DH20, DH21, DH22)
The first object Dumbledore left to Harry was the first Golden Snitch he ever caught during his first Quidditch match at Hogwarts. That was the same match at which Voldemort bewitched his broomstick while possessing Professor Quirrel, but Hermione saved Harry unknowingly by knocking over Quirrell as she hurried to set Snape on fire (PS11, PS17). Scrimgeour told Harry that due to flesh memory, the Snitch would only open for him, and possibly there was something important inside it. It was only after Scrimgeour departed that Harry remembered he caught the Snitch with his mouth. When he pressed his lips to it, an inscription appeared that said "I open at the close" (DH7). The Snitch actually contained the Resurrection Stone from the Peverell Ring that had been one of Voldemort's Horcruxes, and Harry was only able to open the Snitch and use it as he prepared to face Voldemort and die to save his friends (DH34).
The second object Dumbledore left to Harry was the Sword of Gryffindor, but Scrimgeour refused to give it to him, saying it was a historical artifact and not Dumbledore's to give. Snape later gave Bellatrix Lestrange a fake Sword to keep in her vault at Gringotts, and brought the real Sword to Harry in the Forest of Dean, leading him there with his silver doe Patronus (DH19, DH33). After using the Deluminator to find Harry, Ron used the Sword to destroy the Horcrux made from Salazar Slytherin's locket. Dumbledore must have figured out that the Sword was infused with Basilisk venom, one of the few things that can destroy a Horcrux (DH6, DH15), and possibly what he used to release the Resurrection Stone from the Horcrux ring (HBP23).