Sauron, formerly known as Mairon, fears Aragorn for reasons I do not know. I especially do not understand, since Sauron is far more powerful than Aragorn even without the ring, as he is a Maia.
Why does Sauron fear Aragorn?
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There are two main reasons for this:
In addition to the reasons cited in other answers, e.g.,
there is another fear that Sauron had: Sauron does not know who has the Ring. He suspects that Aragorn may have it, and in his pride, is attempting to overthrow Sauron to put himself in Sauron's place.
Aragorn, as a descendant of the Men of Númenor, armed with the One Ring, would be a truly terrifying foe. Presumably, he would be able to control the Nazgûl, Orcs, and Trolls; he could persuade the men of Harad and Rhûn to change sides. Sauron would be left without an army.
It is what he would do in our place. That we would wish to overthrow him and have no one in his place, he has not considered. That we should seek to destroy the Ring has not yet entered his darkest dreams.
(That quotation is from memory; I don't have the books with me.)
Being the rightful King to the thrones of Gondor1 (and Arnor re-established), he had the potential to reunite the sundered Kingdoms of the Dúnedain: Arnor and Gondor, to oppose Sauron.
Then Aragorn, being now the Heir of Isildur, was taken with his mother to dwell in the house of Elrond; and Elrond took the place of his father and came to love him as a son of his own. But he was called Estel, that is "Hope", and his true name and lineage were kept secret at the bidding of Elrond; for the Wise then knew that the Enemy was seeking to discover the Heir of Isildur, if any remained upon earth.
The bigger reason that Sauron was so fearful was that:
'Now Sauron knows all this, and he knows that this precious thing which he lost has been found again; but he does not yet know where it is, or so we hope. And therefore he is now in great doubt. For if we have found this thing, there are some among us with strength enough to wield it. That too he knows. For do I not guess rightly, Aragorn, that you have shown yourself to him in the Stone of Orthanc?'
'I did so ere I rode from the Hornburg,' answered Aragorn. 'I deemed that the time was ripe, and that the Stone had come to me for just such a purpose. It was then ten days since the Ring-bearer went east from Rauros, and the Eye of Sauron, I thought, should be drawn out from his own land. Too seldom has he been challenged since he returned to his Tower. Though if I had foreseen how swift would be his onset in answer, maybe I should not have dared to show myself. Bare time was given me to come to your aid.'
Therefore Sauron had all the more reason to fear Aragorn's power. Though being a mortal Man, Aragorn 'wrenched' the Orthanc-stone against from Sauron's will. That's definitely to be afraid of Aragorn.
I don't think that Aragorn having a very small amount of Maiar blood in him was the reason that Sauron feared him, as many of the other answers given seem to suggest. By the time Aragorn was born, his Maiar blood was so diluted that it can be considered as irrelevant. In this excellent answer Aragorn's Maiar blood percentage is calculated to be
1 : This is perfectly answered here.
Sauron manipulated the Palantíri to deceive, driving Saruman to treason and Denethor to despair, thus nobbling the strongest of his opponents.
Then up pops Aragorn. He has the Palantír of Orthanc, so presumably he has already defeated Saruman while Mordor wasn't looking. What's worse, he has the strength of will to take control of the Palantír, and the clear intention to claim the throne of Gondor, rally its people, and generally do all that Denethor could not.
All that scheming wasted!
I think most of the other answers have overlooked something else. In addition to Sauron fearing him for his potential with the One Ring, which he thinks is Aragorns for the taking, there is this:
He can command men, that is pretty much it. In a fractured world (there are few and weak alliances) he is the one man that can bring them together and put up a fight, even win, especially with the ring.
There are two elements of the answer not mentioned above.
Once a Maia, or even a Vala (such as Morgoth) turns against Eru, they gain no more power. Any power they use is used up, and is lost to them. That is why they can be defeated by those who, originally, had less power than they did. Morgoth is wounded by Shelob and taken captive by Valar once less powerful than himself because of this. Similarly, Sauron lost a lot of power in the fall of Numenor and in making, then losing the One Ring. So he is vulnerable.
Aragorn is the rightful heir of Gondor, the kingdom descended from Numenor, a kingdom of men created by the Valar. This imbues him with power very high among men, close to that of the Maia. So as Sauron descends, unable to replenish his power, and Aragorn ascends, Good prevails over Evil not because it is stronger in this world, but because it is perpetually renewed and reborn from the Source of Life itself.
No doubt Tolkien's view of the triumph of the Good through resurrection of life, which he called "eucatastrophe" is related to his theology, which fits within one of the theories of good and evil within the Catholic doctrine.