I think the "Divine Right of Kings" you mention is probably tangled with modern Protestant, democratic and enlightenment notions about God being a mostly private endeavor and kings being an aberration. Chapter 10 of G.K. Chesteron's "Short History of England" touches on exactly what you want to know*.
He wrote that the supporters of the divine right of kings like Robert Filmer
... professed the impossible ideal of “non-resistance” to any national and legitimate power; though I cannot see that even that was so servile and superstitious as the more modern ideal of “non-resistance” even to a foreign and lawless power.
It's important to understand the difference that may have been apparent to people educated under the Catholic Church in the middle ages that there is a distinction between Natural Law and Human Law.
A king, despot or any other civil authority always has some degree of control over "Human Law" in any society (outside of an actual democracy by mob rule). Who then, do those leaders, whether they're elected or born into the role, gets their authority from? Some would say the people, some would say God, either way it is not "Human Law" which makes a person a ruler, it is a thing natural to man.
Locke would not agree with this, but (according to a Patriot's History of the United States) many of the founding fathers of the United States of America would . Some think government is a natural thing regardless of the form it takes, some think it is a superficial thing external to mankind. In the middle-ages, government was a natural institution.
So, depending who you ask, Divine Law is either a subset or a superset of Natural Law. The Catholic Encyclopedia clarifies it a little bit (hopefully not anachronistically)
This Catholic doctrine concerning the Divine origin of civil authority, as it is inherent in society, must be carefully distinguished from the theory of the Divine right of kings which was popular in England among the High Church party in the seventeenth century. According to the theory of Divine right the king was the Divinely constituted vicegerent of Jesus Christ on earth; he was responsible to God alone for his acts; in the name of God he governed his subjects in both spiritual and temporal matters. The theory united the spiritual and the temporal power in one subject, and derived the combined authority from the direct and immediate delegation of God. It has not ineptly been called Caesaropapism.
So that kind of Divine Right, the kind Henry VIII would have used as a pretense to drain the monasteries of England dry, would not have been the same Divine Right assumed by peasants and rulers alike in the Middle Ages, where submission to the Pope and the Church was a taken for granted.
Furthermore, Chesterton wrote:
It was not really even so simple as this; for the Middle Ages were not, as it is often the fashion to fancy, under a single and steely discipline. They were very controversial and therefore very complex; and it is easy, by isolating items whether about jus divinum or primus inter pares, to maintain that the mediaevals were almost anything; it has been seriously maintained that they were all Germans.
So, I think he's getting at what you are saying as this being a broad question, it's very broad because it means many things.
But it is true that the influence of the Church, though by no means of all the great churchmen, encouraged the sense of a sort of sacrament of government, which was meant to make the monarch terrible and therefore often made the man tyrannical. The disadvantage of such despotism is obvious enough. The precise nature of its advantage must be better understood than it is, not for its own sake so much as for the story we have now to tell.
And the advantage, if not the reason people put up with "Divine Right" might be:
The advantage of “divine right,” or irremovable legitimacy, is this; that there is a limit to the ambitions of the rich. “Roi ne puis;” the royal power, whether it was or was not the power of heaven, was in one respect like the power of heaven. It was not for sale.
Which is very much the same reason Bernie Sanders and Donald Trump have done so good recently, both substantially make the claim that their right to rule would be natural and not for sale (in this case, not for sale to banks instead of barons).
* Short History of England on Gutenberg and Librivox