There used to be a lot of speculation in The Legend of Zelda fan community about how the timeline of all of the games tied together, as some believed that the games existed as separate entities, while others thought that the games shared a timeline that showed glimpses of important moments in the history of Hyrule.
Nintendo finally ended the speculation with the release of Hyrule Historia, which outlined the timeline of the series. According to Hyrule Historia, the timeline broke off into three branches at the end of The Legend of Zelda: Ocarina of Time. The official website for the series and The Legend of Zelda: Encyclopedia have shifted some of the entries around, but the timeline has mostly remained the same.
The Legend of Zelda: Link's Awakening is actually the mid-point of its own series, as it stars the same version of Link from The Legend of Zelda: A Link to the Past, who would go on to star in the two Oracle games. So, how do all of these games tie into each other?
The Fallen Hero Of Time
The Legend of Zelda: A Link to the Past and The Legend of Zelda: Link's Awakening take place in "The Hero is Defeated" timeline, which contains most of the early entries in The Legend of Zelda series. In this timeline, Link is defeated at the end of The Legend of Zelda: Ocarina of Time, leaving Hyrule at the mercy of Ganondorf, who has transformed into a monster called Ganon.
The Imprisoning War
With no one left to oppose him, Ganon gathered all three parts of the Triforce. The exact timeline of the next events is sketchy and has continuity issues, but Ganon essentially transforms the Golden Land into the Dark World and unleashes an army upon Hyrule. The Knights of Hyrule were almost wiped out defending the realm, but they gave the Seven Sages enough time to cast the spell that could seal Ganon away in the Dark World.
Link Defeats Ganon
The events of The Legend of Zelda: A Link to the Past involve a plot to free Ganon from his prison. A sorcerer named Agahnim plots to use the Seven Maidens (the descendants of the Seven Sages) to break the seal and free Ganon from the Dark World.
Link is sent on a quest to stop Agahnim, which involves claiming the Master Sword and freeing the Seven Maidens from their imprisonment. The new version of Link is able to succeed where his predecessor failed and he defeats Ganon. Link then uses the Triforce to undo Ganon's deeds.
Link Sets Sail To Distant Lands
Ganon has been defeated, but Link is unable to enjoy the peaceful era that he has ushered in, so he sets sail for distant lands in order to train his mind and body. Link succeeds in his training and makes his way back to Hyrule, but is stopped along the way by a vicious storm that leaves him shipwrecked on Koholint Island, which sets up the events of The Legend of Zelda: Link's Awakening.
Link was right to keep up his training, as he would go on to have adventures in the lands of Holodrum and Labrynna, as is seen in the Oracle games. Ganon is briefly revived by the sacrifice of Twinrova, but Link defeats him once more and finally brings peace to the land.