Narchuil is one of the lesser Rings, not to be confused with the Rings of Power. These lesser Rings were made as experiments to practice the art of ringforging as taught by Sauron in his disguise.
Before the Shadows of Angmar
Narchuil, the Ring of Truth, was worn by Amarthiel, the champion of Angmar, according to Turbine lore. She lost it however, as she and the armies of the Witch-King were defeated at the second battle of Fornost, by King Eärnur of Gondor and an army of elves led by the elven-lord Glorfindel, who had come too late to aid in the first battle of Fornost, in which the city was overrun by Angmar, effectively destroying the former kingdom of Arthedain, a part of the ancient kingdom Arnor.
Amarthiel was actually Narmeleth, daughter of Laerdan, who had been subverted by Antheron during the Second Age. Laerdan took Narchuil away from her to free his daughter of the influence of Antheron, and he broke the ring into two halves, one of which he kept with him. The other half was given to Aearil, an elf seafarer who went lost with his ship in the icebay of Forochel, where he was looking for the last king of Arthedain, Arvedui.
Shadows of Angmar Storyline
In Volume 1: The Shadows of Angmar, starting with book 9, the player encounters the mysterious woman Sara Oakheart, who turns out to be Amarthiel in disguise. Amarthiel, long supposed dead, steals a palantír from Mordirith, at the moment of his defeat, taking control of the kingdom of Angmar. In book 10, she retreats to Annúminas. From here she starts looking into recovering her old ring, Narchuil. After the player recovers the palantír from her, she moves her army to the Trollshaws, searching ever on in the vale of the river Bruinen, Tâl Bruinen. After the player discovers Amarthiel to actually be Narmeleth, who is possessed of the spirit of the Champion of Angmar. Here he also discovers who the real Sara Oakheart was.
In book 12, the player moves to rescue the elf Laerdan from the clutches of his possessed daughter. He then tells him of the ring-half she took from him and locked away in Barad Dúrgul. After securing the half, the player moves to Forochel, to try and find the ring-half given to Aearil long ago. Eventually the player is able to recover it, which ends book 13.
At the beginning of book 14, Laerdan escapes from Rivendell with the two halves, seeking to find a way to get his daughter back. The player then needs to retrieve some special items to kindle the forges of Mirobel. However, Mordrambor, apparantly in service of Amarthiel again, is faster every time. At last in Eregion, the player confronts Amarthiel with her newly forged ring, and is almost defeated were it not for Mordirith, arriving on a fell beast, to take back his power over Angmar. Amarthiel is struck down, and Mordrambor, having betrayed Amarthiel, is allowed to take the ring from her, as promised by Mordirith. The two then set off to Angmar, leaving Narmeleth, now free of Amarthiel, with one hand missing, as suddenly Laerdan appears, gravely wounded and with his dying breath sees his daughter again as she once was.
Elrond locks Narmeleth away at Delossad, fearing Amarthiel to still hold sway over her. Narmeleth urges the player to help her prove she is free, and eventually succeeds, leading to him and Narmeleth infiltrating Gador Gúlaran, where they at long last defeat Mordrambor, throw Narchuil in the fires, and slay Mordirith, though at the cost of Narmeleth's life.