Breaking The Cypher
Before the war between the humans and the First Civilisation (FC), The FC were weak, and very little in numbers. To prevent the race from dying the FC tried breeding with a humans who gave birth to two children. The babes, however, contained both FC and human DNA, thus a half and half. The twins, who later became Adam and Eve, contained the FC gene and were immune to the mind control affects of the Apple.
When the FC used the Apple to control the humans and provide slave labour for them, Adam and Eve possessed the ability to resist and escape with the Ezio's Apple. The truth video featured in Assassin's Creed II showed Adam and Eve fleeing Eden with the apple whilst being pursued by a member of The FC (Presumably Juno because The Apple was later found in (The Temple of Juno).
Adam and Eve Attempted to free the other humans, but by doing so raging a war between [Those who came before]]. After the war and the humans had reigned victorious; Adam and eve started a new life alongside humankind, unaware their future ancestors would all contain FC DNA. The Apple was passed down through the Generations, falling into the hands of the Templars on occasion, but eventually came in the possession of Desmond in 2012. Desmond, ancestor of Adam, contains the FC DNA (Much like Altair and Ezio) Which allows Him to weild the power of Eagle Vision. There is another, like Desmond who also carries the gene.
When Subject 16 says to Desmond, "Eden, she... in Eden, find Eve. The key, her DNA." He is referring to another human who carries the FC DNA.
It is, In the First Civilisation's eyes, Desmond's destiny to mate with the woman referred to as "Eve" and produce a baby, who shall obtain full FC DNA.
Notes- When Desmond was in Colosseum Vault he stabbed Lucy Stillman, this is suggested to be because The FC thought Desmond was becoming attracted to her, and that she provided a distraction from "Eve".