Towards the end of book/movie 7, Harry becomes the master of all three hallows, thus becoming the "master of death". But what does that mean? Did he become immortal? Or simply that he became unafraid of death?
Using the Harry Potter Wiki as a reference:
While it is generally assumed that becoming the Master of Death refers to some form of immortality, the true Master of Death truly accepts the fact that death is inevitable, and that there are other things worse than dying.
I don't remember the books clarifying the point at all. But then, I also got the impression at the time that
the whole Master of Death thing was not actually true; that the most likely explanation was that the three wizarding brothers had simply created the three powerful artifacts; and that everything in the fairy tale, and the lore about the Master of Death, were simply rumors and misinformation.
In the magical world, for centuries it was assumed that a Master of Death is the one who possesses the three Deathly Hallows, i.e. the one who unites the Elder Wand, the Cloak of Invisibility and the Resurrection Stone.
As quoted by Xenophilius Lovegood in the Seventh book,
"That is a children's tale, told to amuse rather than to instruct. Those of us who understand these matters, however, recognize that the ancient story refers to three objects, or Hallows, which, if united, will make the possessor Master of Death"
"Master or Conquerer or Vanquisher whatever you say"
Possessing the hallows doesn't make one immortal. As in the Tale of the three brothers, nor were the three brothers immortal. It was said that the last thing that could conquer the master of death was Death itself!
However, at the end of the same book, with the small chat with Harry in his dream after Voldemort killed him, Dumbledore clarified the point behind being a true Master of death.
...because the true master does not seek to run away from Death. He must accept that he must die, and understands that there are far, far worse things in the living world than dying.
This was evident from the fact that the third brother who possessed the Invisibility cloak, didn't run away from death, but greeted him as an old friend.
P.S. Harry was never the possessor of all the three hallows during any time at once. Though he had the cloak and the stone, and also was the owner of the Elder wand, he never physically possessed it. Moreover, about the time when he had the Elder wand, he had dropped the resurrection stone in the Forbidden Forest.
The legend was that uniting all three made one "Master of Death". It was, in other words, trying to outsmart Death. For example, the two older brothers chose items that would 'humiliate' Death, not realizing in their arrogance that they were playing right into his hands. Only the 3rd brother asked for a gift that showed wisdom, seeking to evade Death, but the request showed reverence, even respect, in that only Death's own cloak could work against Death himself.
An individual who owned or won the gifts, but, did not lose to them could essentially be considered a Master of Death.
The Wand: Harry won the wand and used it to defeat Voldemort - but, he didn't want it. He used it to repair the phoenix wand. Therefore, Harry was master over the Elder wand as it did not hold him sway. He gave it back.
The Cloak: Belongs to Harry as it belonged to his father before him. Harry inherited the Cloak, therefore is its master.
The Stone: The real proof of Harry's mastery of Death lay here. Harry took the stone and used it not to bring others back from the dead, but, to have them give him the courage to face his own death. Dumbledore insisted that Harry didn't realize how special he was because he could love. Harry, in this very instance, shows the greatest possible love by laying down his own life. This is how Harry masters the stone, because he uses it to give him the strength to allow Voldemort to slay him, thereby breaking the curse (destroying the piece of Voldemort's soul that had attached to Harry as a child, thereby making him the seventh horcrux) and proving the prophecy (...he shall have power the Dark Lord knows not). Because Harry used the stone to willingly face Death - the very thing that Voldemort feared - Harry proves that he is Master of Death. Additional proof of Harry's mastery over the stone is that Harry did not keep the stone, either. He discarded it in the forest after his loved ones escorted him to his destination.
The scenes with the horcruxes, however, are not the only ones in which Harry shows that he has an equal footing with death. In PS, Harry defiantly tells Ron and Hermione that Voldemort with have to kill him, because he was never joining the dark side. In CoS, after the basilisk has bitten him and he is slumped against the stone wall, Harry muses that '...if this is dying, it's not all that bad...) [paraphrase] In GoF, when readying himself to fight Voldemort, Harry decides that he is going to die fighting, standing tall like his father. In OotP, when Voldemort attempts to possess Harry in the hope that Dumbledore will sacrifice Harry in order to kill him, Harry, suffering from the pain of the connection with Voldemort, mentally begs Dumbledore to kill them, reflecting that death would be better than this pain and that he, Harry, would be able to see Sirius again. In HBP, in his hatred and sorrow, Harry screams at Snape, calling him a coward and dares Snape to kill him like he killed Dumbledore.
Throughout the entire series, after he learns how his parents truly died, Harry continuously proves that he has been prepared to face death all along. He does not willingly seek death; he is not suicidal. But, he does not fear death and he understands, as Dumbledore stated, that there are far worse things than death - something that Voldemort never understands. This makes Harry a Master of Death.