On the show, the Night King is the imposing leader of the White Walkers. While his appearance is based on the White Walkers from the books, the show has added much of its own invention to this character. In the books, there is a character named the Night's King, but he is mostly unrelated to the White Walkers. The Night's King was the 13th Lord Commander of the Night's Watch—before he deserted his post to pursue a woman who was as cold as ice, "with skin as white as the moon and eyes like blue stars."
He sets up his own dominion in the Nightfort, which he ruled as a king for 13 years. During his reign, horrible atrocities were committed, and many men of the Night's Watch died. It wasn't until the King in the North joined forces with the King-Beyond-The-Wall that the Night's King was finally defeated. Additionally, in the books the White Walkers do not appear to have any noticeable social structure or hierarchy, let alone a king. So other than the name, there's no real connection to the Night King as seen on the show.