Taylor Swift: A young singer's meteoric rise

Music sensation Taylor Swift writes and sings all her own songs, sells millions of records and she's just 21 years old

She was a pioneer in using social media to connect personally with her fans: posting funny video blogs she edits herself with glimpses of her offstage life, making her fans feel like they're part of her close circle of friends.

And she's orchestrated her concerts too - to get as close to her fans as any performer we've ever seen. Halfway through the show she walks through the audience and sings three songs to the people in the back.

All while members of her team search the crowd for the most enthusiastic fans, and reward them with gold! An invitation to hang out with Taylor after the show.

Then Taylor heads back to the stage through the crowd. Touching and hugging all over again and when the crowd roars, her expression of awe - again and again - can be, well, hard to believe.

Lesley Stahl: Are you really surprised, or are you just kind of putting it on?

Taylor Swift: I'm really surprised every time I see a crowd like that. 'Cause I never thought I'd get to play to a crowd like that.

Lesley Stahl: So when you go [imitates her making face], it's real?

Taylor Swift: Does it look like that? Great.

[Excerpt from "Thug Story" video]

One of the things her fans love about her is that she laughs at herself, as in this video with rapper T-Pain...

[Taylor Swift: I knit sweaters, yo!]

Poking fun at her squeaky clean image... and turning her uncoolness into cool!

[Taylor Swift: You guys bleeped me and I didn't' even swear.

T-Pain: she didn't even swear.]

Taylor Swift has won just about every music award there is, including the industry's highest honor: the Grammy for Album of the Year in 2010.

But the few setbacks in her meteoric career have come, ironically, on those award nights. As when Kanye West grabbed the mike from her and the time she sang a live duet with Stevie Nicks at the Grammies... off key. One nasty review said she had killed her career overnight, and was "too young and dumb to understand the mistake she'd made."

Taylor Swift: The things that were said about me by this dude, just floored me and like leveled me. And I-- I don't have thick skin. I hate reading criticisms. Like you never-- you never really like get past things hurting you.

But then Taylor did her thing, and turned the wound into a song, the hit single "Mean."

And in the music video Taylor broadened it beyond herself - to a boy in a locker room reading a fashion magazine, a girl who shows up wearing something different. The song has taken on a life of its own - a sort of anti-bullying, anti-meanness anthem.

There's a deep deep connection here. As one of her fans told us: "Taylor lets us know it's okay to be ourselves."

Lesley Stahl: Scott Borchetta says, 'She's a cultural leader and she knows it.'

Taylor Swift: Well, I definitely think about a million people when I'm getting dressed in the morning. That's just part of my life now. And--

Lesley Stahl: You're a role model and you know it.

Taylor Swift: I think it's my responsibility to know it, and to be conscious of it. And it would be really easy to say-- you know, I'm-- I'm 21 now. I do what I want. You raise your kids. But it's, that's not the truth of it. The truth of it is that every singer out there with songs on the radio is raising the next generation. So make your words count.

Lesley Stahl: What is it like to achieve your dream so early?

Taylor Swift: You know-- it's great.

Lesley Stahl: The answer is it's great.

Taylor Swift: You know, it's not bad. And it just means that I have a lot of time to figure out how I'm going to prove myself over and over and over again and-- and I have time to do it.