Rose in 2x08 explained to Elena that she was the key to break the curse put on Klaus.
Rose: Because you are a Petrova doppelganger. You’re the key into breaking the curse.
Elena: The curse? The sun and the moon curse?
Rose: Oh, you do know your history.
Elena: What do you mean I’m the key? The moonstone is what breaks the curse.
Rose: No, the moonstone is what binds the curse. The sacrifice will breaks it.
Elena: The sacrifice?
Rose: The blood of the doppelganger. You are the doppelganger. Which means, in order to break the curse you are the one who has to die.
It was later confirmed by Katherine in 2x09:
Katherine: The curse was bind by the sacrifice of Petrova blood. Witches are crafty with their spells. The doppelganger was created as a way to be able to undo the spell. Once the doppelganger reappeared, the curse can be broken.
In 3x13 Elijah explained that Esther also used Tatia’s blood to make them vampires both (besides using it to put the curse on Klaus):
Elijah: Our mother was a very powerful witch. She sought to end our feud over Tatia, and so she took her. And Klaus and I would later learn that it was Tatia’s blood that we consumed in the wine on the night where our mother performed the spell which turned us into vampires. Tatia wouldn’t make a decision between the two of us, so for a time, Niklaus and I… Grew estranged. Harsh words were traded. We even came to blows.
As Elijah explained in 2x19, The Sun and The Moon curse doesn’t exist. Klaus and Elijah made that up. The only curse that existed was the Hybrid curse put on Klaus:
Elena: I don’t understand. So Klaus drew the Aztec sketches about the curse?
Elijah: Roman scrolls, African tribal etchings, and any other culture or continent we felt like planting in it.
Elena: But why?
Elijah: Easiest way to discover the existence of a doppelganger or to get your hands on some long, lost moon stone is to have every single member of two warring species on the lookout.
Elena: So it’s not Aztec at all?
Elijah: The curse of the sun and the Moon… is fake. It doesn’t exist.
Elena: So Klaus’ real father was from a werewolf bloodline? What does that make Klaus? A werewolf? Or a vampire?
Elijah: He’s both. A hybrid would be deadlier than any werewolf or vampire. Nature would not stand for such an imbalance of power. Therefore the witches, the servants of nature, saw to it that my brother’s werewolf side would become dormant.
Elena: That’s the curse that Klaus wants to break?
Elijah: He wants to trigger that part of him that’s a werewolf. If allowed, Klaus would sire his own bloodline. He’d build his own race. Endangering not just vampires, but everyone.
We know that the Originals are a thousand years old, so they lived during the 10th-11th century. We know Katherine was born towards the end of the 15th century, and Elena was born during the 20th century. Klaus is trying to break the curse since the moment it was put on him. So he looked for the doppelganger for centuries and we know how crafted he is. He had to wait around 500 years for Katherine to be born, and then again another 500 years for Elena to be born. So there’s this repeat of the 500 years that occurs.