When I was watching TV coverage of the Obama inauguration I saw the usual stars and stripes alongside another flag which looked almost the same except it had stripes with a fewer stars in the box and arranged in a circle. I know way back when the US flag was like this. What was the flag and why would it have been rolled out on this occasion?

13 stars in a circle - that would've been the Betsy Ross, the first flag of the United States.

http://en.wikipedia.org/wiki/Betsy_Ross_flag

So when and why is it used?

Google ;Betsy Ross

I did. I couldn't find the answer...just the history.

The setup is explained here:

http://media.newsobserver.com/conten...auguration.pdf

Why exactly it's flown along with the other historic flags? I guess to show the history of the United States, how it came from 13 original colonies to 50 states?

Sort of like the Six Flags Over Texas, but that's a whole different story.

Ah, I though you meant this one...

Which should not be confused with this one...the state flag of Hawaii...