The Link in The Legend of Zelda and The Adventure of Link is the first Link appearing in the The Legend of Zelda series, and the third Link chronologically in the Downfall Timeline.
In the original The Legend of Zelda, Link is a young swordsman who is asked after rescuing Impa, the royal nursemaid, to save Hyrule and Princess Zelda from the evil Prince of Darkness, Ganon, who is seeking out the shards of the Triforce of Wisdom. Link collects the pieces of the Triforce of Wisdom and enters Ganon's lair in Death Mountain. He defeats Ganon with a Silver Arrow, secures the Triforce of Power, and saves the princess from Ganon's clutches.
In the sequel Adventure of Link, set six years after The Legend of Zelda, Link is now fifteen years old, approaching his sixteenth birthday. Although Link has defeated Ganon, the remnants of Ganon's army are still scattered across Hyrule. They plan to revive Ganon by spilling the blood of his murderer, Link, onto his ashes. Impa finds a Triforce mark on Link's hand as he approaches his sixteenth birthday, and explains that he is the hero chosen to awaken an ancient Princess Zelda from a potent sleeping curse by seeking out the Triforce of Courage. This time, Link places a crystal in each of Hyrule's six palaces, ultimately destroys Dark Link and receives the Triforce of Courage in the Great Palace, and awakens Princess Zelda.
Since Link's met many of his predecessors and other Links, when around them, he's called 'Brunette' to stand out from the rest of them. The nickname comes from the fact that he's the only Link with brown hair.