The Night King was the first of the White Walkers. He was one of the First Men before he was turned into something else by the Children of the Forest.
She said some people believe Night's King was a Bolton, a Magnar of Skagos, an Umber, a Flint, a Norrey, or a Woodfoot. However, she identifies Night's King as a Stark of Winterfell and brother to the King of Winter and suggests his name was Brandon.
No, the Night King is not a Targaryen.
In the books, however, the story of the Night King is slightly different. The Night King isn't the leader of the White Walkers. Instead, he's a legendary historical figure who supposedly betrayed the Night's Watch to marry a “corpse queen” and then crowned himself a king.
The Night King was created by the Children of the Forest during the war of the First Men and the Children of the Forest and remained alive for about 10,000 years. It can be assumed the magic that transformed him also gave him immortality.
In Bran's vision, he saw how the Children of the Forest created the Night King: by driving a blade of dragonglass into the chest of a human who'd been bound to a weirwood tree. (It's unclear whether the blade also penetrated the weirwood tree itself, but it seems likely.)
In Season 8, Bran reveals that the Night King wants to destroy him because he holds all of the history of man.
King Aerys II the Mad King, son of Jaehaerys II and great-nephew of Aerion, he is arguably the most infamous monarch in the Targaryen dynasty overall.
According to Old Nan, the Night's King may in fact be (hold on to your small clothes) a Stark! If that isn't enough, legend has it that the Night's King was not just any Stark, he was King Brandon Stark's brother—or possibly Brandon Stark himself!
Based on that history, there's no way the Night King can be a Targaryen. He would have had to have been a captive member of the race of the First Men.
As the Three-Eyed Raven, Bran has the ability of greensight, which means he can see events in the past, present, and future through visions. Killing Bran would essentially erase all memory of mankind. Samwell Tarly explains it pretty well in the scene.
TL;DR version: The Night's King is interested in Jon Snow because he sees some of himself in the current Lord Commander. “The gathering gloom put Bran in mind of another of Old Nan's stories, the tale of Night's King.
It wasn't Jon. Nor Daenerys. Instead, Arya got the hero's moment at the end of the Battle of Winterfell. After thousands fell during the fight in “The Long Night,” Arya managed to kill the Night King — and destroy every single White Walker and wight in the process.
So, it shouldn't be a shock to anyone that he was able to withstand the direct blaze from Drogon's fire. And his immunity to fire doesn't have anything to do with a possible Targaryen lineage but rather everything to do with his magic. The only way to defeat him is with Valyrian steel, and that's just what Arya does.
No, Hodor is not a Targaryen.
In the TV series, it's revealed that Hodor was originally a stable hand called Wylis who was essentially lobotomized when Bran attempted to warg into Wylis, connecting him with present-day Hodor who was holding back an army of wights.
So the basic thrust here is that, according to fan theory, the current Starks may be descended from White Walkers, thanks to a child born of a human man and a White Walker woman known to legend as the Night's King and the Night's Queen respectively.
Aerys's son, Rhaegar Targaryen — Daenerys's brother, the crown prince and heir to the throne, whom we now also know as Jon Snow's father — fell in love with Lyanna Stark, Ned Stark's sister, even though he was already married to Elia Martell.
The Origins of the Night King
We have no idea who he was before he was turned into the Night King, but we can be sure that he wasn't a Targaryen. During the Children of the Forest's conflict against the First Men (which they were losing) they created the first White Walker from one of the First Men they had captured.
Aegon IV Targaryen, known as Aegon the Unworthy, was the eleventh Targaryen to sit the Iron Throne, and is considered to be one of the worst Targaryen kings. He sired numerous bastards, legitimizing them on his deathbed, an act that led to five Blackfyre Rebellions.
Joffrey Baratheon is arguably the most hated character in Game of Thrones. As the sadistic and cruel King of the Seven Kingdoms, he delights in inflicting pain and suffering on those around him, including his own family members.
Before he died of old age in 94 AC (After Conquest), Balerion was known throughout the Seven Kingdoms as the most terrifying creature alive, and he was no doubt the most fearsome asset to House Targaryen's rule.
All of that changed when Bran entered the picture as the Three-Eyed Raven. This introduction gave the Night King a true motive outside the typical megalomania of Big Bads. By wiping out the Three-Eyed Raven, he would be wiping out all memories of the previous world and fully ushering in his new era.
Game of Thrones season 2 shockingly revealed that the Wilding Craster gave up his male children — born of incestuous relationships with his own daughters — as sacrifices to the Walkers in exchange for relative peace in the Haunted Forest.
During the Battle of Winterfell, twelve White Walkers enter Winterfell along with the Night King to kill Bran Stark. All the White Walkers were destroyed after Arya killed the Night King.