If there is one iconic feature that defines Harry Potter, it would be his scar (his Lily-green eyes coming a close second.) He is told that the lightning-shaped mark on his head came from the same car accident that killed both his parents, a lie believed until Hagrid shows up at 4, Privet Drive with a completely new story.
Harry learns that he's the chosen one, that he survived the evilest Dark Wizard of the 20th century while just a baby, and that it is his destiny to face Lord Voldemort in the end. The scar plays an important role in his hero's journey, but there are several confusing aspects that are not really explained in the narrative.
Why Is It A Lightning Bolt?
The scar is not shaped like a lightning bolt for any specific reason, and besides, nobody in magical history had ever survived a direct hit from the Killing Curse before, and there is, therefore, no knowledge of what the resultant scar might look like.
More importantly, it seems to be an artifact of Voldemort's accidental Horcrux-creation, and not directly related to Avada Kedavra. Rowling herself mentioned that her design choice was "because it's a cool shape. I couldn't have my hero sport a doughnut-shaped scar."
How Is It Related To His Mother's Sacrifice?
As the blood magic between Harry and his mother is triggered when she becomes a living shield against Voldemort, the Dark Lord's curse is mirrored: destroying his corporeal form and leaving Harry completely unscathed.
Well, obviously not completely, because it leaves a gash on his forehead that scabs into the famous scar. This is very interesting, because why would Lily's death not offer her son absolute protection? Are there some sacrificial love loopholes that haven't been considered yet?
It Is Linked To Voldemort's Horcrux
Voldemort's soul is split once again when he attempts to murder an innocent baby, although it's actually 1/12th of his total soul (because he's already made six Horcruxes at that point.)
Nevertheless, the piece inadvertently embedded itself inside little Harry, making him the final secret Horcrux. However, a combination of Avada Kedavra and sacrificial protection is clearly not the standard method for making Horcruxes, so how did that even happen?
The Scar Is A Portal
The author claims that the scar hurts because there's "this piece of soul really wanting to get back out the way it entered", which means that it forms a skin-based doorway for the exchange of spiritual essence.
In other words, every time Voldemort is around, or if his emotional state turns extreme, the Horcrux tries to dig its way out of poor Harry's head. It's still not clear if the soul fragment is physically attached to the scar, or if it tumbles around aimlessly until drawn to its source.
Why Doesn't Voldemort Sense The Eighth Horcrux?
Continuing from the above argument, if the eighth Horcrux within Harry tries its best to return to Voldemort, this means that it experiences what the Dark Lord does. Along the same lines, this should also mean that he should be able to perceive its existence, but he evidently has no clue that his soul is shattered more times than he had planned.
Voldemort can access Harry's mind through their connection, sending him visions of Sirius being tortured, but he somehow cannot comprehend why they share such a link in the first place?
What About Harry's Early Years?
After failing to kill his "chosen" nemesis, Voldemort flees into the wild Albanian forests, hoping that one of his minions would seek him out. Of course, this doesn't occur, so he spends most of his time possessing random animals to survive (mostly snakes), only escaping his misery when Quirrell comes along on one of his expeditions.
And for eleven years, Harry's scar didn't make a single sound? Was it because Voldemort was half a world away? Or because the details of their connection hadn't been fleshed out yet?
Why Isn't There A Second Scar?
Harry willingly walks into the Forbidden Forest to accept his fate at the hands of Voldemort and is completely unsurprised when he is struck by his infamous jet of green light. This acted in the same way as his mother's sacrifice, and permanently prevents the Dark Lord from hurting anyone at Hogwarts (probably even the world.)
The attack unwittingly destroys his own soul, but Harry comes back from the dead with no further injuries, but such a thing is only possible if the Killing Curse was aimed at the scar with pinpoint accuracy. If not, where's the new lightning bolt?
Why Couldn't Harry Use It To Read Voldemort's Mind?
Voldemort happily utilized their bond to cause great psychological distress for Harry, that too when he is at his teenage angstiest. Dumbledore isn't successful at getting him to learn Occlumency, though, because his feud with Snape reaches boiling point during their lessons.
Could he have learned Legilimency, instead, and rifled through the Dark Lord's innermost thoughts? Harry can certainly feel his emotions, and even see through his eyes, but not perform some basic telepathy?
It Actually Hurts Him After 19 Years
The final words of The Deathly Hallows mention that the lightning scar had not caused him any kind of pain in 19 years, strongly suggesting that Harry would never have to deal with Voldemort anymore.
And yet, in The Cursed Child, when Delphini "Riddle" uses time-turners to restore her father back to his rightful place, Harry experiences a burning once-familiar feeling in his forehead. Voldemort is technically dead at this point, so why on earth would it reactivate?
Does It Ever Disappear?
The scar is resistant to any kind of healing magic, irrespective of how powerful, because of its dark origins. Which is fine, but shouldn't it vanish like any other non-magical scar after the whole situation is resolved? It remains on Harry's forehead, regardless, although it does fade a considerable amount over time.
There is no Horcrux trying to squeeze its way out anymore, so the portal should have been fully sealed. Then again, how can something so recognizable be made to disappear?