This Princess Zelda is one of her latest incarnations, making her only appearance in The Legend of Zelda: The Minish Cap. Instead of being a royal and rarely-seen princess of earlier games, in the beggining she's childhood friends with Link and even walks freely around town. But she still gets kidnapped once again, if you can call getting turned to stone that.
So Link must defeat the evil Minish-turned-monster Vaati who started the trouble to begin with.
Then you can restore things to normal. Even though she's a bit forgettable, I do like this Princess Zelda's back-story as a nice change of pace.