The Phantom Menace first mentions the prophecy of the one who would bring Balance to The Force. Later episodes begin throwing the idea of destroying the Sith into that.
Then Yoda says in Episode III "A prophecy that mis interpreted could have been", and indeed it was, while what Obi-Won says to Anakin after he defeats him on Mustafar implies he thought the Prophecy had failed, most SW fans agree Anakin's turn was part of the Prophecy's plan all along, or at least always an option for it.
Where I differ from most is that most do seem to be under the impression that the prophecy wasn't fully fulfilled until Return of The Jedi. It is interpreted as being that bringing Balance to The Force required destroying both orders. And I myself had fallen into that trap in the past.
The conclusion I have come to is that the Prophecy is never discussed in the Original Trilogy because it is irrelevant to that Trilogy. It was 100% fulfilled by the time the credits rolled at the end of Episode III.
We are never shown exactly what the Prophecy says in the movies, I don't know or care if any EU work ever does. The Sith are not connected to the Prophecy at all when we're introduced to it. I frankly think the Sith were inserted into how it was interpreted possibly between Episodes I and II (in universe) simply because the Sith returned at the same time this Chosen One was found.
Common sense says that when we have hundreds or thousands of Jedi and only two Sith, that the dis-balance is in the Jedi's favor therefore it was their order that needed to be destroyed. But they blinded themselves to the possibility that they themselves were the problem.
When Revenge of The Sith ends we have exactly two Sith and two Jedi left, and two New Born force sensitives who's fates are undecided. I know EU material skews those numbers, but going off the actual Six Episodes that is how it was perfectly balance when the Prequels ended.
One of the Criticisms of the Prequels is that it was Jedi overkill, but that was intentional. The order is larger then it was ever meant to be.
Through out the Prequels the Jedi are not what they were meant to be. The non Military aspects of their function became forgotten most of the time. And by Episode III they were headed into dark territory with the situation they were forced into.
But even before we entered Episode I, the fact that they take children away from their parents and raise them to be soldier monks at such a young age is a major ethical problem, yet even in Episode V Yoda is still thinking that was Ok, that Luke is too old, when he should have realized starting Anakin's training so young was the problem, and Luke's age was in fact the ideal age to start.
The Jedi Order blamed the Dark Side clouding things for their inability to use it properly. But was that really the sole reason? Maybe they should have considered that they'd spread the Light Side way too thin.
Prequel haters are a lot like Gene Roddenberry. Roddenberry hated that the Star Trek stories people actually liked don't fit his idealized Utopic vision of the Federation. Prequel haters likewise hate that the Jedi order we saw didn't fit the idealized view of it they got from how the OT seemed to describe them. The difference is, with Star Wars the Prequels were the creator's true vision all along, but the simpler story was easier to introduce people to the world. Not unlike how the late Third Age of Middle Earth was a lot simpler then the First Age.
Obi-Won and Yoda were remembering things through rose tinted glasses. And people forget even going off the OT alone they were WRONG about what needed to be done to defeat the Sith. Luke went against their advice in trying to redeem to his Father and succeeded. They wanted Luke to simply fight and kill both Vader and The Emperor, and there was really no way he could have done that. So if they can be wrong about the future, they can be wrong about the past.
Return of The Jedi if anything was an undoing of what the Prophecy achieved, now things are out of balance again. How does this effect The Force Awakens? For one thing I think the Knights of Ren being created to fill the void left by the Sith's destruction was inevitable.
When the title was first announced people complained "The Force doesn't Sleep", But I've been thinking, since the Dark Side was always more active, the Jedi being more purely reactive, perhaps in a sense The Sith being destroyed did make it become somewhat dormant, and now the Knights of Ren are waking it back up.