Divine Right in the modern context is the power to rule given by God, however the concept of Monarchs who had been 'chosen' and with amazing powers existed before Christianity came to Europe - Celtic Kings (and Queens) were regarded as divine, and the concept also existed in Egypt, China and Japan - in China this was taken to the extreme degree that King actually
was a God, not just that he had been given the gift of rulership. The concept was pretty much thrown out in the UK after the Civil War and the growing authority of Parliament, however you can still see echoes of that concept even today - see the Princess Di effect for proof!
And again you only have to look at the Princess Di effect/phenomenon to see how people even now have folk belief that Kings (or Princesses) can 'heal' them.
EDIT (now I've got some dinner

) : It's also worth considering how Tolkien uses the concept. Aragorn indeed has the right to be King, but he does not exercise this right with force. The idea of Divine Right has some
very negative connotations, including the arrognace displayed by Charles I, and by some of the French Bourbon kings (resulting in the bloody Revolution). Note that Aragorn (interestingly with that name which is so close to the word
arrogant) does not abuse this right. He
almost does, at Meduseld, but Gandalf holds him back and makes him consider his actions. Instead, Aragorn very much earns his Kingship through example and leadership, and displays humility when his 'secret' is discovered via Ioreth's knowledge of folklore.
Contrast this with Denethor who thinks he has some kind of 'divine right of Stewardship'. I think Tolkien very much shows that though Kings may indeed have been bestowed with this 'gift', they still must earn the moral right to put it into practice, much as a democratically elected leader might.