There are so many things that helped make Game of Thrones one of the greatest television shows of all time, from the character arcs to the political drama to the human stories, and the fantasy elements that got thread throughout its eight seasons deserves praise also. Dragons were the primary source of this, but the most significant aspect of fantasy for a long time was the White Walkers and the Night King.
For years fans debated whether or not the Night King was the real big bad of the show or if it would all come down to a battle for the Iron Throne. Across the entire show, we learned a lot about the Night King, perhaps not enough for a lot of fans, but a decent amount.
A First Man
About twelve thousand years before the events of the series, the original human inhabitants of Westeros were the First Men, a race that invaded the lands and ruled it for thousands of years before the invasion of the Andals from Essos.
Before the Night King was the icy villain fans came to know, he was one of these Westeros inhabitants who, likely, did battle alongside the other First Men against the Children of the Forest.
Captured By The Children Of The Forest
Speaking of the Children of the Forest, they were the original inhabitants of Westeros and were a non-human race who existed for years, but as of the events of the show, appear to be extinct, dying with Leaf finally; however, there could still be a few more around.
The Children of the Forrest were at war with the First Men when the latter invaded, and in the process fo that, they captured the soon to be Night King and turned him into what he became.
How & Why He Got Turned
The Children of the Forest did not necessarily create the Night King and White Walkers deliberately, but they got desperate and were experimenting with their magic to protect themselves from the First Men, a plan that ultimately backfired.
The way they managed to do this by way of their magic, and when the captured the man, they drove a dragonglass dagger into his heart, resulting in his eyes turning blue, giving way to the first White Walker, the Night King.
The Long Night
As mentioned, though, this plan backfired. The White Walkers eventually turned on the Children of the Forest, and the White Walkers went on a course of destruction across Westeros.
This happened during the longest winter in recorded Westerosi history, the Long Night; it lasted a generation, leaving people dying of either hunger or the White Walkers.
His Retreat
However, the Night King and the White Walkers did not come out of this Long Night the victors or rulers of Westeros, with a war happening in which they lost and got forced to retreat.
That was the War of the Dawn, fought between the White Walkers and the newfound alliance between the First Men and the Children of the Forest. The White Walkers got drove back to the Land of Always Winter, with the Wall subsequently getting built, and the Night King and his army falling into obscurity and, later, legend.
Special Powers
The Night King is a character with a variety of special powers, making him the ultimate fantastical threat to the people of Westeros during the time of the show.
These include Cryokinesis (the ability to freeze anything), necromancy (raising people from the dead), White Walker conversion, superhero-like strength, the ability to mark people, so he knows where they are at all times, enhanced durability, seismic control, the power to shatter weapons and objects, sensing wargs, fire immunity, natural dragon-riding, prolonged life, weather manipulation, and likely more than fans are aware.
His Army
Throughout his life, the Night King, alongside his fellow White Walkers, accumulated a colossal army that would eventually appear at the Battle of Winterfell; this was the Army of the Dead.
The Night King, and the other White Walkers, all brought animals and people back from the grave, everything from Giants to humans to dragons. The Army of the Dead, a.k.a Wights were extremely susceptible to fire, and extremely expendable, but were also large in number, all falling when the Night King fell.
The Books
Like so many things, there is a vast difference between the Night King of the series and the Night King in the books thus far, with two novels still scheduled for release from George R. R. Martin.
In the books, the Others are mentioned very little, and as for the Night King, he is yet to have a proper book counterpart. This leaves a lot of things up in the air regarding the next two books and how they will match up to what happens in the show.
His Goals?
Combining the two worlds of the series and the novels is eventually going to be necessary to understand the Night King, as the series really does not make clear what exactly he wants, save for Bran's word on the matter, and did not flesh him out enough.
Bran says the Night King wants to see the world in eternal winter and get rid of the humans who currently rule it, erasing the world and its memory. This idea is good but could have gotten explored more, but perhaps The Winds of Winter and A Dream of Spring will do so.