Link is the hero of the The Legend of Zelda, and its many sequels, prequels, and it's-really-not-clear-quels. He is not Zelda! Zelda is a girl's name, and Link is male. Also, he wears a skirt or tights most of the time.
Since there seems to be no end to the clods getting Link and Zelda confused, we'll explain it: The series is referred to as The Legend of Zelda (sometimes shortened to just "Zelda", but never shortened to just "The") because as foretold by the gods of Hyrule, every so often in the world's history, its legendary princess, Zelda, will be reborn, and traumatic events will threaten her and her kingdom. "The Legend of Zelda" refers to how she always is fated to be reborn.
Except wait; so is Link. And also Ganon. And some other guys like Tingle, because the gods of Hyrule hate us. Screw this explanation; it's your fault; Nintendo.