If Harry Potter was himself a horcrux, then why did Voldemort die even though Harry was alive? I understand that Voldemort's spell-"Avda kedavra" was ineffective against Harry, because of the Elder Wand. Did that short period death (visit to King's Cross Station) of Harry fulfill the criteria to destroy the horcrux that resided in his soul? If the horcrux was inside his soul then how is it possible to destroy part of Harry's soul when he had never killed anyone like Voldemort did?
As Dumbledore explains in the movie:
On the night Lord Voldemort went to Godric's Hollow to kill Harry and Lily Potter cast herself between them the curse rebounded.
When that happened, a part of Voldemort's soul latched itself onto the only living thing it could find: Harry himself. There 's a reason Harry can speak with snakes.
There's a reason he can look into Lord Voldemort's mind.
A part of Voldemort lives inside him.
That part (latched to Harry, not his soul) was killed by Voldemort himself:
Harry: Professor, what is that?
Dumbledore: Something beyond either of our help. A part of Voldemort sent here to die.
Knowing what we know now, it makes sense that only this tiny piece of Voldemort soul was killed: the Elder Wand refused to hurt its master, so Harry was unharmed (although, given a choice to move on), but that part of Voldemort's soul that made Harry a horcrux had nothing to do with the wand's master, so it was killed.
Voldemort dies because all 7 Horcruxes are destroyed. Harry became one of them, when Voldemort attempts to kill Harry on the night Lily and James Potter die. When Harry got the scar he also got powers. A little piece of Voldemort lived inside of him. That's why Harry can speak Parseltongue and can see Voldemort's thoughts.
After Harry, Ron and Hermione destroy the other Horcruxes (and Neville kills Nagini), Harry duels with Voldemort. The Elder Wand goes to Harry because Draco Malfoy had disarmed Dumbeldore, and Harry had then taken Malfoy's wand.
Some have mentioned that Harry survived because the Elder Wand cannot hurt its master, but I think the actual cause is deeper. Harry survived the killing curse because at that time, he possessed all of the Deathly Hallows, and thus was "Master of Death".
- Harry owned the Cloak of Invisibility, and had for some time.
- Harry was master of the Elder Wand. Though he did not personally posses it, it's clear it served his will, and not Voldemort's.
- Harry was the last possessor of the Resurrection Stone. Though technically he has already tossed it away, I believe that like the Elder wand, the stone would still view Harry as it's master even if it was not physically on his person.
So as "Master of Death", Harry simply could not be killed. Even the powerful killing curse that sent him to the White King's Cross could not, ultimately, completely send him on if he still had the will to live.
On the other hand, the part of Voldemort's soul that lived in Harry had no such protection, and was predictably slaughtered by Voldemort's own curse. Thus the Horcrux dies and Harry survives.