The Westron name for Dúnadan was simply Adûn, "westerner", but this name was seldom used. This name was reserved to those Númenóreans who were friendly to the Elves: the other, hostile survivors of the Downfall were known as the Black Númenóreans.
Sauron's spirit fled from Númenor to Middle-earth, and he again raised mighty armies to challenge the new Dúnedain kingdoms, Gondor and Arnor. With the aid of Gil-galad and the Elves, Sauron was defeated, and he vanished into the wild East for many centuries. Gondor and Arnor prospered during this time.
As Sauron began to re-form and gather strength, a series of deadly plagues came from the East. These tended to strike harder in the North than the South, and caused a population decline in Arnor. Also the chief of the Nine Ringwraiths, known commonly as the Witch-king of Angmar, began assaulting the divided Northern Dúnedain kingdoms from a mountain stronghold (Carn Dûm). Eventually, he succeeded in destroying Arthedain, the last of the Northern kingdoms.
After the fall of Arthedain, a remnant of the northern Dúnedain became the Rangers of the North, doing what they could to keep the peace in the near-empty lands of their Fathers. The surviving Dúnedain population of Arnor retreated to the Angle south of Rivendell, while smaller populations made isolated settlements in far western Eriador.
Over the centuries, the southern Dúnedain of Gondor intermarried more and more with so-called Middle Men. Only in regions such as Dol Amroth did their bloodline remain pure. Their lifespan became shorter with each generation. Eventually, even the Kings of Gondor married non-Dúnedain women occasionally.
In the Fourth Age, the Dúnedain of Gondor and Arnor were reunited under King Aragorn II Elessar (who was also called the Dúnadan). He married Arwen, daughter of Elrond (a cousin many times removed, as it were) and reintroduced Elf-blood into his family line.
In addition to the Faithful, there were Dúnedain in the South who manned Númenórean garrisons at places like Umbar. Many of these folk had been turned toward evil by Sauron's teachings, and remained loyal to him after the fall of their homeland. These are referred to as the Black Númenóreans.
In addition, Dúnedain, especially those of high rank, possessed great wisdom and discernment, and occasional prophecy. They benefited from longer life-spans (three times the life of a regular man) than ordinary men and could retain their youth until the very end of their days. Though the reason is not fully explained in the 'Tale of the Years', one thing that led to the diminishment of the Dúnedain was their tendency to have few children, in many cases only one. Aragorn came from a long line of only children.