Abstergo has taken 17 different descendants of assassins to experiment on and use their memories to find the Pieces of Eden, remnants of a forgotten society that have the power to control people. Abstergo is an agent of the Templars, who want to use the Pieces to rob people of their free will and create eternal stability. The Assassins are the order that has tried to stop them. The battle has raged on throughout history.
Desmond Miles, the main character of Assassin's Creed I and II, is Subject 17. Before him, 16 subjects failed to produce anything much of value. Subject 16 was left in the Animus too long, which killed him. He suffered an overdose of the bleeding effect, which made him unable to tell who he was, what time he was in, or where he was at any given moment.
However, before he died, he was able to hack Abstergo's systems and obtain a wealth of knowledge about the Templars and the Pieces of Eden. He hacked into the Animus and put the knowledge in glyphs to make it useful to the next descendant of an Assassin who was put in the Animus.
It turns out that the Roman gods such as Jupiter and Minerva, were all part of an advanced society that existed before the time of humanity. This society, the first civilization, was called Eden. This society was destroyed, as is explained at the end of the game, by a shift in the sun's magnetic poles, much like people are talking about right now in real life for 2012. The advanced people of Eden lived in harmony with humanity, providing them with the advanced technology, hence the city and the suits Adam and Eve are wearing in this video.
In this video, Adam and Eve are taking a Piece of Eden, an apple, and running away with it. This is symbolic of the 'first sin' wherein Adam and Eve take the apple from the Garden of Eden.
Adam and Eve are humans in this advanced society, which is kept in peace by using the Apple of Eden. There is no free will. The Templars later have the same aim: to use the Apple and the other Pieces of Eden to control people and make peace by manipulation. Adam and Eve are against this in their society, and they were the first humans. This is why, in the last Abstergo email, they are talking about how the truth about humanity's origins makes them sick. Humanity is descended from Adam and Eve, who shared the goals of the Assassins: free will; not the goals of the Templar. To be descended from early assassins disgusts them.
The reason Assassins are chosen for the Animus and not the average human is because a) Assassins are closer descendants of Adam and Eve than most, hence why they can move like they do, and b) Assassins and their families are more likely to have experience with the pieces of Eden than the average person.
The first civilization was destroyed by a massive solar flare, similar to the ones predicted for 2012 in reality. Minerva speaks to Desmond at the end of the game, telling him, through Ezio, that the same thing is going to happen again if the two most powerful factions in the world, the Templars and Assassins, don't break their obsession with the pieces. Minerva is likely one of the members of the ruling caste who abused the Pieces, but she helps Ezio so the Templars and Assassins will not make the same mistake they did.
Adam, Eve, and the humans (seen in the factor laboring for the ruling caste) survive the destruction of the world, along with the Pieces of Eden. The 'gods' become nothign more than legends. The Pieces were built by the rulers to manipulate the humans they lived with into being peaceful, hence why modern humans are so susceptible: the Pieces were built for them. It is also said that Jesus, another Biblical figure, used a Piece of Eden to manipulate his followers, hinting again that the mythical people of Eden were abusers of the Piece, and Templars.
In AC1 and AC2, Desmond is being used by Abstergo to try and discover the locations of the Pieces without killing him, hence why his Animus time in AC1 was so segmented: they didn't want him to die before he could get the truth, which is what happened with 16. This also would prevent him from learning any of Altair's fighting abilities and escaping. In AC2, they let him stay in so his memories begin to bleed. This is how he learns his fighting skills, and also how he sees Altair conceiving a child with Maria. If he stayed in longer, he would have gained more knowledge of his history, at the cost of his life. Theoretically, a close enough descendant of Adam could go back as him in the Animus.
It is also mentioned in the puzzles that the Piece of Eden is the connection between apes and humans: apes evolved because of the Edenites' technology, and then the world was destroyed, yielding humans; to cover this up, Abstergo planted caveman skeletons in Africa.