Alfred Borden (Christian Bale) is actually two twin brothers who use the same name to be disguised in order to pursue their career as magicians.
Robert Angier (Hugh Jackman) wanted to get his revenge from Borden for killing his wife by accident. He did not put in mind that she wanted Borden to tie her up differently for the trick.
Tesla created a machine for Angier to help him with his magic tricks. However, the machine did not work as intended; as it clones the user instead of transporting him.
It is not revealed whether Cutter knew about Borden from the beginning or whether he was told later on about the siblings. At the end of the movie, it is revealed that he knew that there were two Bordens and helped at least one of them.
The idea of seeking justice began to vanish in Angier's head when he sought to beat Borden with his fame.
Angier became a killer. He did not care whether it was himself that he killed or his clone. Even Tesla did not know which one was the original and which one wasn't.
One of the Borden brothers sacrificed himself so that the daughter wouldn't grow up without a father. The other Borden is the one who killed Angier.
One of the Borden brothers was always disguised. Nobody knew about them. The two people who came closest were Sarah and Olivia. The brothers took part in wearing the disguise through the period of the movie.
Borden's trick was very simple as there were two Bordens who are identical twins. Angier couldn't understand it because one of them had a cut-off finger. What happened is that the other Borden cut off his own finger as well.
Borden's diary was nothing but a piece of scam to fool Angier. He knew all along that Angier sent Olivia for him.
Olivia fell in love with the other Borden who wasn't married. When the other brother's wife saw that, she became jealous because nobody could tell them apart or knew of their existance. That is why she betrayed Angier, despite Borden not trusting her.