Why are some elves royalty?

Question

In Tolkien's work there are some elves who are kings, ladies or lords. Elrond is the Lord of Rivendell, Galadriel is the Lady of Lorien and Turgon was the King of Gondolin.

They are described as either the founders of their realms or children of kings. But realms are not built singlehandedly and someone must have become a king originally.

All elves are descended from a group of equals that awoke in Middle Earth. They had no internecine wars apart from three Kinslayings and they have not competed for territory or resources. Yet many of their domains have rulers and nobility while some immortals are servants for eternity.

By what right do some elves rule over others?

Answer 1

Elf royalty comes from the order in which they woke, proximity and exposure to the Valar, and through descent from other elves who were already established as rulers.

The elves did not all wake up at the same time. The first three elves that woke up became the leaders of the first three elf houses.

While their first bodies were being made from the 'flesh of Arda' the Quendi slept 'in the womb of the Earth', beneath the green sward, and awoke when they were full-grown. But the First Elves (also called the Unbegotten, or the Eru-begotten) did not all wake together. Eru had so ordained that each should lie beside his or her 'destined spouse'. But three Elves awoke first of all, and they were elf-men, for elf-men are more strong in body and more eager and adventurous in strange places. These three Elf-fathers are named in the ancient tales Imin, Tata, and Enel. They awoke in that order, but with little time between each; and from them, say the Eldar, the words for one, two, and three were made: the oldest of all numerals.
_{"The legend of the Awaking of the Quendi" (The War of the Jewels)}

The rulers that are in place at the beginning of the Silmarillion (Ingwë, Finwë, Elwë/Thingol) are related via direct patrilineal descent from those first three leaders, though Tolkien was undecided how many generations apart they were.

Ingwë, Finwë, Elwë, grandsons of Imin, Tata, and Enel
_{Draft of "Suggestions for key dates" (The Nature of Middle-earth)}

Let Ingwë, Finwë, Elwë all be young 6th gen. Elves, but each a direct descendant (by eldest son) of Imin, Tata, and Enel [respectively]. (Divergence in dates of birth is due to intrusion of earlier-born daughters.)
_{Generational Schemes (The Nature of Middle-earth)}

These three elves were furthermore then brought to Valinor as ambassadors of their race. They thus became the first elves to experience the light.

Therefore Orome was sent again to them, and he chose from among them three ambassadors; and he brought them to Valmar. These were Ingwe and Finwe and Elwe, who after were kings of the Three Kindreds of the Eldar; and coming they were filled with awe by the glory and majesty of the Valar and desired greatly the light and splendour of the Trees.
_{"Of the Coming of the Elves" LQ§23 (Morgoth's Ring)}

All of the Noldor ruling line, (e.g. Fëanor, Fingolfin, Turgon, etc) are directly descended from Finwë. During the second age, Gil-galad, as the last surviving member of this line in Middle-earth was acknowledged as high king over all the remaining elves.

In Lindon north of the Lune dwelt Gil-galad, last heir of the kings of the Noldor in exile. He was acknowledged as High King of the Elves of the West.
_{"The Tale of Years" (The Lord of the Rings, Appendix B)}

Around that same time many of the Sindar elves traveled eastward and colonized realms among the Silvan elves that lived there. Celeborn and Galadriel weren't necessary Sindar, but their realm fits the same format.

In the beginning of this age many of the High Elves still remained. Most of these dwelt in Lindon west of the Ered Luin; but before the building of the Barad-dûr many of the Sindar passed eastward, and some established realms in the forests far away, where their people were mostly Silvan Elves. Thranduil, king in the north of Greenwood the Great, was one of these.
_{"The Tale of Years" (The Lord of the Rings, Appendix B)}

Other rulers come from other means. Elrond is ruler of Rivendell because he was in command of the military force that established the stronghold.

When news of this reached Gil-galad he sent out a force under Elrond Half-elven; but Elrond had far to go, and Sauron turned north and made at once for Eregion. ... Elrond had gathered such few of the Elves of Eregion as had escaped ... but he was forced away northwards, and it was at that time that he established a refuge and stronghold at Imladris (Rivendell).
_{"Concerning Galadriel and Celeborn" (Unfinished Tales)}

Answer 2

In the passage of the Similarillion which deals with the Valar's first encounter with the Elves, it describes the three kindreds of the Eldar - Vanyar, Noldor and Teleri. At this point those kindreds already have lords - Ingwë, Finwë, Elwë, and Olwë (the Teleri had two lords as there were more of them).

Shortly before that, we read that after Oromë the Vala encountered the Elves at Cuiviénen, he chose three of them to take to Valinor so that they would convince the rest to journey there. According to the Similarillion:

He chose from among them ambassadors who should go to Valinor and speak for their people; and these were Ingwë, Finwë, and Elwë, who afterwards were kings.

So it seems that the concept of "lordship" existed among the Elves from very early on, and was shortly elevated to "kingship" thanks to the choice of the Valar.

And this is not surprising, given Tolkien's general outlook. In his worldview, some people simply are kings or lords, and that is their deserved place.

(And as I've said before, he was not in any sense a historian and was not at all interested in how kings and rulers developed in the real world.)