Just to pad out and extend loopydude’s excellent exposition (and because I like to hear myself type):
Elendil and his sons, Ilsildur and Anarion sailed away from the wreck of Numenor. Elendil was the last lord of the Andunië, a royal line descended from Elros, brother of Elrond and first king of Numenor.
Elendil landed in the north, at the Grey Havens by Cirdan and Gil-Galad the Elven king. There he founded the northern realm of Arnor.
Isildur and Anarion landed in the south, by the Pelagir where many of the “faithful” of Numenor had fled during their oppression by the king of Numenor. These people were loyal to the Valar and to the lord of the Andunië. There Isildur and Anarion founded the realm of Gondor. Isildur’s city was Minas Ithil (later Minas Morgul) and Anarion’s city was Minas Anor (later Minas Tirith). Together they ruled in Osgiliath, where they had thrones set side by side.
But they acknowledged their father Elendil as the High King of both Arnor and Gondor and of all peoples of the west. When Elendil died in battle with Sauron, Isildur (being the elder son) inherited this title. His brother Anarion had died during the Siege of the Barad-dur. Isildur stayed in Gondor for a time, instructing his nephew Meneldil in the arts of ruling, then went north to take up the scepter of Annuminas, token of the High King.
But Isildur never made it north. He was slain, along with his 3 eldest sons. It fell to his young son Valandil to eventually take the scepter of Annuminas. But Valandil never tried to exercise lordship over Gondor and ruled as king of Arnor only, not as High King. The two kingdoms remained allied, but each had their own ruler.
Indeed, no king of Arnor (or Arthedain, it’s successor state) ever tried to claim the kingship of Gondor until Arvedui Last-King did so, upon the death of Ondoher, I think. He was married to Ondoher’s daughter, Firiel. And he raised the issue of the High Kingship, stating that he was the holder of that title.
But the nobles of Gondor rejected that claim, and they appointed Earnil king. He was also of the line of Anarion. Arvedui accepted their decision, and even sent to Earnil for help when the north kingdom began to founder. Help came, albeit too late to save Arvedui. It did help his son, though.
It was only with Aragorn that the High Kingship was again taken up. Notably by a descendant of both Isildur (from father to son unbroken) and Anarion (via Firiel, daughter of Ondoher and wife of Arvedui).
JRRT implies that there were other descendants of Elendil possibly still out there. After the kin-strife of Gondor their numbers were reduced by murder, but many packed up and headed to the hinterlands, or renounced their claims. But most of these lines were interwoven with those of non-numenorians, and considered to be less pure. Tho one notable king of Gondor was only half-numenorian and did ok for himself.
Even so, there were probably a few mortal descendants of Luthien still banging around, outside of the House of Elessar Telcontar.