The Story - A Link to the Past - The Land of Hyrule

To understand the story of A Link to the Past one must first understand the legend of the Triforce (according to ancient Hylian scrolls). First, the world was created by three gods. The God of Power created the land and the mountains. The God of Wisdon created science and nature. The God of Courage created all life. With their work done, the gods departed, but not before creating a symbol of their strength - the Triforce. The Triforce is a small golden triangle made up of three smaller triangles. It was left in the Golden Land where it called to people hoping someone worthy of its three titles (The Forger of Strength, The Keeper of Knowledge, and the Juror of Courage) would find it. Over time, as the legend of the Triforce was passed down from generation to generation, it became distorted.
One thing that kept the legend alive, however, was the Book of Mudora, a collection of old Hylian legends. In it was a passage about the Triforce:

In a realm beyond sight,
The sky shines gold, not blue.
There, the Triforce's might
Makes mortal dreams come true.

The promise of having one's wish granted kept people searching for the Triforce. It was eventually found by a gang of thieves skilled in black magic. The leader of the gang, Ganondorf Dragmire, made his way to the Triforce and had his wish granted.
No one knows exactly what Ganondorf wished for, but as a result of his wish, greedy men were drawn to his army. The sky became darkened, and Hyrule suffered many disasters. The Seven Wise Men and the Knights of Hyrule were sent to seal the entrance to the Golden Land in an attempt to stop the flow of the Triforce's power. A sword was forged that was able to resist evil magic even if it was granted by the Triforce. This sword is known as the Master Sword. However, it could only be wielded by someone with a pure heart and a strong body. The Seven Wise Men continued to search for the one who could wield the sword, but soon Hyrule Castle was attacked by Ganondorf and his men. Many died in the battle, but the Seven Wise Men were able to seal Ganondorf in the Golden Land, thus stopping the evil flow of power. This is what became known as the Imprisoning War.
Centuries later, as the Imprisoning War became a faded memory, Hyrule was once again ravaged by pestilence and drought. The seal on the Golden Land still held, so the king offered a reward to anyone who could find the source of Hyrule's troubles. A wizard named Agahnim came along and using his magic, brought Hyrule back to its peaceful state. He was given the title of chief advisor and heir to the Seven Wise Men. However, rumors had been circulating that Agahnim was not the hero people saw him as. There were stories of strange magical experiments in the castle tower at night. The people once again lived in fear.
One night, Link was awakened by a girl's voice communicating with him telepathically. She was asking for help. She told him her name was Zelda. Link got out of bed in time to see his uncle leaving. He said he'd be back by morning. And so begins one of Link's greatest adventures.

A Link to the Past is the game that really started the puzzle-solving aspect of the Zelda series. It flawlessly combined that aspect with the exploration element that Zelda fans had come to love. In short, this game is the basis of all modern Zelda games. That said, if you haven't played this game, do whatever you can to get your hands on it. If you're still not convinced, read on.
A Link to the Past is set in a huge world - two huge worlds, actually. After obtaining certain items, Link can travel between the two worlds at will. The first world is the Light World. This is Link's world where he has lived his entire life (it's the "normal" Hyrule). The second world is the Dark World, the world created when Ganondorf's wish was granted. At times, to solve certain puzzles or reach certain areas, Link will have to travel through both worlds. He will also collect a wide variety of items, some of which are exclusive to this game, some of which are now classic Zelda items. Some of the exclusive items include the bug-catching net, the Book of Mudora, and the three spells. Some classic items are the hookshot, the bow and arrow, and the boomerang. There are also a wide variety of side-quests and minigames which can result in more health, more magic, more bottles, and item and sword upgrades.
Many Zelda fans including myself consider this the greatest 2D Zelda game ever created, and one of the best games in the series. To anyone looking to play a really great game, don't pass this one up.