This is the Link from the The Legend of Zelda, The Adventure of Link and The Legend of Zelda: Breath of the Wild. He was born to Arn and Medilia in the country of Calatia. Around this time he was nearly 16 years old Link left home and travelled to the land of Hyrule, a land being oppressed by an evil monster.
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Once he arrived (in The Legend of Zelda) there he heard from Impa, Princess Zelda's nursemaid that Zelda had been kidnapped by Ganon. He travelled around the land collecting the Triforce of Wisdom shards that Zelda had spread around Hyrule to keep them safe from Ganon, who already had the Triforce of Power. Link eventually defeated Ganon and saved Princess Zelda, and now claimed two-thirds of the Triforce. He stayed in Hyrule and helped them rebuild. When he saves Princess Zelda she dubs him Hero of Hyrule.
Later, (in The Adventure of Link) on his 16th birthday his hand started to glow with the Hyrule Royal Family crest. Years ago a prince misused dark magic and accidentally cursed his sister into an eternal slumber while trying to get the Triforce of Wisdom from her. His sister was the original Princess Zelda. Her father placed her in a sealed room in the Hyrule castle and cast a spell on the land of Hyrule that would make the family crest become visible on the hand of a boy who contained the virtue, age, experience, morality and personality necessary to harness the power of the Triforce. Only will that boy have the strength of heart to use the Triforce's power to break the curse on Princess Zelda, rather than use the wish for himself. The prince, to make sure the tragedy was never forgotten, ordered that all future princesses in the Hyrule family be named Zelda.
Link showed his crest to Impa. She took him to a door in the Hyrule Castle that no one had been able to open. When she pressed his hand against the door it unlocked, inside it was a chamber where the original Princess Zelda slept. Inside the room were six crystals and a scroll from the King, that explained why he had the crest and instructed him to place the crystals in several shrines across Hyrule. Placing these crystals would break the magical barrier around the Great Palace in the Valley of Death where the Triforce of Courage rests.
Meanwhile, Ganon's minions were gradually recouping and planning to resurrect him. If Link should spill his blood, it could be poured over Ganon's ashes and used to revive him. Should Link be defeated, Ganon would return.
Link eventually defeats Ganon's minions and the King's guardians in each palace, and places each stone, ultimately breaking the seal. He as the final challenge to prove his worth, he defeats his doppelganger, Dark Link, and the withered King presents him with the Triforce of Courage and disappears. Link unites the Triforce and wishes to wake Princess Zelda.