The details of the brothers' disagreements are nowhere recorded, but their result was of historical significance, causing the old North-kingdom of the Dúnedain to be split into three realms. Amlaith's realm was reduced to the lands west of the River Baranduin and the Weather Hills, a region that came to be known as Arthedain. The eastern lands of Arnor were divided between Amlaith's brothers, creating the new kingdoms of Rhudaur to the north and Cardolan to the south.
Whatever the cause of the dissension between Amlaith and his brothers, the fact of his direct descent from Isildur was never in question. Through Amlaith, the Kings of Arthedain carried the bloodline of Elendil down through the generations.
The date of Amlaith's birth appears only in The History of Middle-earth volume XII, The Peoples of Middle-earth. It cannot therefore be considered completely reliable.
The meaning of Amlaith's name is extremely obscure. The initial am- element is probably 'up', but -laith is much more difficult to interpret. The only other appearance of this element is in the name Limlaith, an Elvish name for the River Limlight, so on this rather shaky basis we can translate laith as 'light'. Why Amlaith should choose to name himself 'up-light' is a mystery - perhaps it hints at a lost story of Amlaith's youth, or perhaps Tolkien was simply lighting his pipe as he thought up the name! Whatever the reason, it is now lost to us.
Amlaith's younger brothers (whose names were frustratingly never recorded) were the founders of the kingdoms of Cardolan and Rhudaur. It should be noted that there is no specific evidence that Amlaith had only two brothers, and indeed there is the vaguest of hints in The Peoples of Middle-earth that he may have had more, but if so they left no mark on the history of Middle-earth.