"We tend to talk more in policy terms and we've got our 10-point plans, but we don't have a narrative, I think, of what it means to be a Democrat," he tells The Early Show co-anchor Hannah Storm. Obama, keynote speaker at this year's Democratic National Convention in Boston, was elected in November to the U.S. Senate.
"I think that if you ask the average person on the street: 'What does it mean to be a Republican?' - whether they agree with the Republicans or not, they have a clear sense of what the Republican Party stands for, and we're going to have to, I think, do some intellectual work to make sure we understand what it means to be a Democrat."
His unusual journey is chronicled in his autobiography, "Dreams From My Father," which he wrote almost a decade ago after he graduated from Harvard Law School. It has just been reissued with the inclusion of the speech he delivered to the Democratic convention this summer.
In the book, he talks about his drug and alcohol use as a young man.
"I went through a period when I was a teenager, where I was rebelling against everything, and I think embracing a lot of the exaggerated stereotypes of what it means to be an African-American. But, fortunately, I had some wonderful mentors and teachers who, I think, pulled me out of it. I had the love of my mother to stabilize me. And part of the reason I wrote about that period in my life in the book was to make clear that there are all kinds of young African-American males out there who are as talented as I am, as energetic as I am, but also as confused and they may not have the same margins of error that I did."
When Obama was a few years old, his black African father left his white American mother to pursue his PhD at Harvard, and never returned. Obama was raised in a white household.
"I think that all of us in modern America are trying to figure our identities because we've got these colliding cultures all of the time, and very few of us stay in one place," he says. "Our entire lives, we're moving around. Obviously, here in New York City, it's the epitome of that kind of hybridization. But I think when it happens along the racial lines, I think it's especially tough."
When Obama is sworn in as a senator, he will be only the fifth African-American senator in U.S. history, and the sole African American in the current Senate.