The original OoT storyline released by Nintendo.

Link is preparing for his coming-of-age ceremony, in which he will receive his guardian spirit. In the tradition of his people, children receive a fairy from the Fairy Tree when they reach adulthood. This fairy becomes a person's lifelong familiar. The fairy accompanies the youthful adventurer seeking his or her fortune in the maze-like forest or in the lands beyond.
Link's ceremony, however, is not destined to be a happy one. The Fairy Tree, source of guardian spirits, is captured by a strange creature from the depths of the forest. Link is able to locate his guardian spirit, Navie, and with her help, slay the creature. However, in the process, the Fairy Tree is grievously damaged. As its life force ebbs, the Fairy Tree speaks the words that will shape Link's destiny.
"Do not allow the thief, Gannondorf, to claim the Triforce ... " the Tree
mystically communicates. "Oh brave one, you must take this sacred stone to a wise man...."
Gannondorf was infamous throughout the land for his evil practices as the king of thieves. He lusted for the power of the Triforce and searched throughout Hyrule for its resting place, most recently plunging into the forest of Link's people. Link knew he was in grave danger should Gannondorf find him. He took the sacred stone from the Fairy Tree and set out for Hyrule Castle, the capital of the Hyrulian people.
Upon entering the castle, Link is welcomed by the young Princess Zelda, a woman of the Hyrulian royal family about the same age as he. She is well aware of the crisis facing the land of Hyrule. Zelda relates further details about the Triforce's hiding place, telling Link that he needs to find the three sacred stones that fit into a magical Ocarina, which serves as the key to hidden realm.
Link's race with the evil Gannondorf to find the sacred stones and the hiding place of the Triforce is on!

Zelda 64 tells the story of an earlier age of Hyrule, when Gannon was merely known as the king of thieves and not the powerful creature of evil he becomes in the other Legend of Zelda adventures.