The story elaborates on what is told of these characters in the published Silmarillion, starting with the childhood of Túrin, continuing through the captivity of his father in the Nirnaeth Arnoediad, and Túrin's exile in Doriath, to Túrin's time in Nargothrond, his unintentionally incestuous relationship with his sister Nienor, and ultimately ending with suicide by his sword Gurthang after having slain Glaurung.
The Narn (as it is commonly called) is a long story of all that happened to Húrin and his children Túrin Turambar and Nienor, after Húrin was cursed by Morgoth. In the Silmarillion, a complete but less detailed version of this story appears as "Of Túrin Turambar".
As a point of reference regarding the names of the main characters: In this story, Túrin renames himself Turambar, meaning Master of Doom in the High-Elven speech, with a vow to turn aside from the darkness that had ruled his early life. His sister Nienor is also called Níniel, meaning Maid of Tears. She is renamed by Turambar himself after he finds her alone and in distress in the woods. Only much later does he learn her real name and origins. The section ends with the Elves calling them by the names of Túrin Turambar Dagnir Glaurunga (as the slayer of the Dragon Glaurung) and Nienor Níniel.
The story has some inconsistencies when compared with the Silmarillion, and at points there are gaps and multiple versions: this is because Tolkien never really finished the story during his lifetime, and his son Christopher Tolkien had to choose from all the work to create a consistent narrative for the Silmarillion.
The story of the Narn continues in the Later Narn, which is also published in the Unfinished Tales, and in the Wanderings of Húrin, a text which was found to be too different in style from rest of the Silmarillion, but which continues the Narn past Túrin's death and ends with Húrin's eventual release and the bad deeds which come from that. This story was finally published in The War of the Jewels, a part of the History of Middle-earth series. A book that was released in 2007 incorporated all the elements of this story. it was called The Children of Húrin.
Note that the title of the Narn is given as Narn i Hîn Húrin in the published Silmarillion and Unfinished Tales. This was an editorial decision by Christopher Tolkien which he later regretted, done only to prevent people from pronouncing Chîn like English "Chin".