Dori, Nori and Ori were all descendants of the House of Durin, though not from the direct royal line that led to Thorin. Their relationship is not explained in the canonical texts, but notes reproduced in The History of the Hobbit reveal that Dori and Nori were brothers, and that Ori was their cousin.
One of the thirteen Dwarves of the Blue Mountains who set out to reclaim the Lonely Mountain from Smaug, under the leadership of Thorin Oakenshield. Nori was the brother of Dori, and the two shared the colour of their purple hoods. Both Nori and Dori were capable flautists, and seem to have carried their flutes with them. Both these apparent brothers also shared Bilbo's liking for regular meals.
Nori did little to distinguish himself during the Dwarves' adventures in the Wild, though he shared the same array of experiences as the others of the Company: captured in turn by Trolls, Goblins, Spiders and Elves, they eventually reached the distant Lonely Mountain and recovered it for the Longbeards. Afterwards he remained in the Kingdom under the Mountain, and Glóin reported that he was still there seventy-seven years later, at the time of the War of the Ring.
Notes
1
Fíli and Kíli were known to be the youngest of Thorin's companions by some fifty years. The elder brother, Fíli, was born in III 2859, so Nori must have been born at least fifty years before that date.
Like many other Dwarves in Tolkien's work, Nori's name comes originally from the Old Norse poem Völuspá. The translation is not certain: various sources suggest different interpretations, including 'little scrap', 'turner', or 'shipper'. Given that Dori has a name that seems to mean 'borer', then 'turner' probably makes most sense in context.