what was the mistake made by US President Barack Obama in his inauguration that requires him to repeat it for the second time..

He was not required to repeat it. Technically he was president before taking the oath of office. He chose to repeat it because the person administering the oath mixed up a few words:

http://ac360.blogs.cnn.com/2009/01/2...roberts-obama/

it doesnt matter, what matters most is what he's going to do next