It was a straighforward dictionary hack--her password was "popcorn".
To reset a Yahoo! Mail password, you need the person's birthdate, zip code, and answer to their "secret question". That information is easily accessible for public figures like Palin. Try it sometime with your friend's email/screenname and Facebook; it's quite easy.
Where I work, all webmail is blocked, so the IA department is grateful to Palin for the object lesson supporting the policy.