Warning: This entire answer is one big spoiler. If, unlike the OP, that is not what you want, don't read it.
Assassins Creed 2 began with Desmond's escape from Abstergo (the Templars), along with Lucy. After escaping, they meet a group of modern day Assassins, with their own, improved Animus. They ask Desmond to use their Animus to relieve the memories of Ezio Auditore, an Italian assassin to find information on the pieces of Eden, and to improve Desmond's skill as an assassin.
Ezio's part in the plot begins with the execution of his family as part of a conflict between nobles that is related to the Assassins/Templar conflict. Ezio discovers his father's past as an assassin, and on the influence of his uncle Mario, takes his father's place. During this point, the main villain Rodrigo Borgia, becomes obvious. When Ezio finally confronts Borgia, he discovers that Borgia has one of the pieces of Eden, which he then takes.
Ezio's allies to this point reveal they are all Assassins, and that they believe Ezio is a prophet. They take him into their ranks.
The next two sections of the game are DLC only, so I will skip these as many players will not have played them.
When you return to Ezio, he is much older (now with his distinctive beard), Borgia has become Pope. Ezio runs through a level in Rome, and finally comes to the Pope in the middle of giving a Mass. He tries to kill Borgia stealthily, and for a moment it appears successful, but then you find out that the Papal Staff is another Piece of Eden. Ezio loses this fight, and when he comes around, you have to follow Borgia for a little while, and fight him once more. This time he does not have the papal staff as he has used it to open a special chamber. Ezio also discards his weapon, and you fight hand to hand. Ezio declines to kill Borgia, as he does not see any gain from it.
Finally you enter the chamber that Borgia had been trying to enter, finding a hologram of a god-like being called Minerva, which explains of how those beings and humankind had fought for a long time, but following a global catastrophe worked together to rebuild the world, but her people still died out. You discover that she was speaking directly to Desmond, through Ezio.
At this point, Abstergo finds the Assassins, and they have to flee, and the game ends.
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Best Answer
I can't find the sources but from what i remember of my game play, the contracts with the diamonds are random and keep respawing, the ones with the circle actually have sort of a mini-story in the description and don't respawn.
So in essence you can't finish them all