Link is a traditional name in Hyrule which has endured the test of time. It is a name which conveys strength, wisdom, and courage, and it is possibly no coincidence that many who have averted major disaster in Hyrule have shared a lineage - and the name Link.

Link, Hero of Time

Zelda 5 (Ocarina of Time)
Zelda 6 (Majora's Mask)

This particular Link was adopted by the Kokiri when his dying mother sought the refuge of the Great Deku Tree after being marauded by the minions of Ganondorf. He was chastised as a youth by the Kokiri (who thought he was one of their own), for he was different from them in not having a guardian fairy. But when cursed by Ganondorf, the Great Deku Tree assigned Navi to be Link's guardian; after the Tree's death, Navi and Link became close companions.

Ganondorf tricked Zelda (the Hylian princess) and Link into opening up the Sacred Realm, during the process of which, Link's mind was frozen in stasis while seven years passed. During these seven years, Ganondorf used his strength and magic to completely destroy the kingdom of Hylia and wreak havoc over all of Hyrule. After Link awoke, however, he soon acquired the ability to travel between the two disparate realities - his childhood and adulthood - in order to smite Ganondorf.

And in doing so, created Ganon.

Although Link, Zelda, and the six sages combined could not destroy Ganon, as he held the triforce of power, they were able to seal him within the sacred realm. And Link, Hero of Time, returned to his childhood to live it out in full.

There is another legend of this Link in which he saves a world which is fated to be crushed under its own moon, but the history is incomplete on this account. (i.e. I haven't beat Zelda 6 yet. :)

Link, Guardian of Hyrule

Zelda 3 (A Link to the Past)
Zelda 4 (Link's Awakening)

Several thousand years after the fall of Hylia, a distant descendant of the Hero of Time met with his own piece of destiny. Although he was raised as an ordinary peasant living on the outskirts of Kakariko, one night he heard a distressful message in his mind; the princess Zelda had been imprisoned by Aghamim, the wizard aide to her father the King, and was to be sacrificed, as were supposed descendents of the sages (although they were, strangely enough, all of Hylian blood, and not the motley crew of varying races which the original sages were; perhaps when their respective races died off, only the original Zelda - who was, in fact, the seventh sage - managed to procreate, in such a way that there were presently seven female descendents or something; this is one of the few places where the histories are in conflict).

This Link would eventually recover the Triforce of Power from Ganon, save Hyrule, go off on a voyage, get shipwrecked, save Koholint (the land of dreams), return to Hyrule, marry the princess, and have children, who would in turn have children, one of whom would have a child, named after his great grandfather...

Link, Defeater of Ganon

Zelda 1 (The Legend of Zelda)
Zelda 2 (The Adventure of Link)

It is unknown how Ganon managed to recover the Triforce of Power, but he did. Using its energy, he returned to Hyrule and destroyed most of its civilization (including, sadly, Kakariko). After many years of his tyrranical destruction, his thirst for power reintensified - the triforce of power was no longer enough. In his utter rage, he kidnapped the princess Zelda, who was in possession of the Triforces of Wisdom and Courage. In order to save what remained of Hyrule, she split the Triforce of Courage into eight pieces, to be sought out by this Link so he could gain the courage to defeat Ganon once and for all.

Which he did, and life was good.

And Hyrule could then begin to rebuild. But all was not right.

Upon Ganon's death, his minions immediately became enraged with a bloodlust for Link, as they knew that by killing him and spreading his blood on the ashes of Ganon, their strong leader would be resurrected. But this wasn't all - an evil presence mysteriously appeared on this Link's birthday, putting Zelda into a deep sleep and causing much chaos across the land. And so Link sought out to destroy this evil, which turned out to be his own shadow, having been turned evil by the sheer power of the Triforce. But this shadow was defeated, Zelda was awoken...

...and they lived happily ever after.