A central part in many of the Legend of Zelda games, this artifact is also referred to as "the golden power", and is the embodiment of the three goddesses that created the world of Hyrule. Its final resting place is at the location where the goddesses departed Hyrule. The Triforce is actually an assemblage of three lessor Triforce pieces, each with its own unique power.
Triforce of Power
The Triforce of Power is the top triangle in the Triforce, and is usually associated with Ganon. It is the essence of the goddess Din, who created the realm where Hyrule exists. When its powers are bestowed upon an individual it typically gives the holder immense magical powers and strength.
Triforce of Wisdom
Located at the lower left of the completed Triforce, the Triforce of Wisdom is the essence of Nayru; creator of the physical rules that bind all things. It belongs to Zelda and seems to enhance Zelda's own magical ability. She has typically displayed telepathy, precognition, and astral projection, along with other magical abilities. It is not fully known how many of these abilities are natively hers and which have been imbued by the Triforce.
Triforce of Courage
Typically associated with Link, the Triforce of Courage is located on the bottom right of the completed Triforce. This segment is associated with, and contains the energies of the goddess Farore, who was responsible for the creation of all life in Link's world. The powers of the Triforce of courage are not fully known, though it seems that it can protect its wielder from certain magical effects. Most often though, it is used in prophecy to designate a hero who will save Hyrule.
The completed Triforce is said to grant one wish to whomever touches it. This has resulted in villians seeking the completed Triforce for their own, nefarious gains.
Ganon, the most well known of these evildoers, has repeatedly tried and failed at wishing on the Triforce to become the ruler of Hyrule.
The Triforce first appears in the original Legend of Zelda on the NES, but it isn't called the Triforce here, it is simply described as "Magical Golden Triangles". In the original game the Triforce of Wisdom is split into eight fragments, spread out over eight dungeons for our hero to collect, and to use on the final 9th dungeon against the archenemy Ganon.
During the second game, Zelda II: The Adventures of Link, the Triforce of Courage lies in the Great Palace, and serves as a remedy for the curse that a wizard has put on Zelda, in order to gain access to the palace, Link must venture into 6 different dungeons and place sacred crystals.
In the third iteration of the Zelda franchise, The Legend of Zelda: A Link to the Past, the Triforce is to be found in an alternate world, called the Golden Land, and legends tell that whoever touches the Triforce will have their greatest wishes fulfilled. After capturing seven maidens, including Princess Zelda, that are descendants of the Seven Sages that captured Ganon during the Imprisoning War, the Wizard Agahnim opens a portal to the Golden Land in order to release Ganon to once again to reign terror on the world, in doing so the Golden Land is turned into a dark horrid land where evil rules supreme by touching the Triforce.
In Ocarina of Time, we learn that the three goddesses, Din, Nayru and Farore create the world at the beginning of time, and when they see that everything is as it should be, they transform themselves in the powerful Triforce in the Sacred Realm. Once again our villain Ganondorf, the King of Thieves and Ganon in disguise, seeks out the Triforce in order to rule the world, and with the unwilling help of Link and Zelda, his plans comes together. After being locked in the Sacred Realm for seven years, Link is released upon the world to save it from Ganondorf, by combining his Triforce of Courage with Zelda's Triforce of Wisdom.
Although the Triforce itself is never spoken of by name, the symbol is frequently seen in The Legend of Zelda: Phantom Hourglass. Throughout the game, Link collects Power, Wisdom, and Courage "Force Gems" to level-up each of the fairies that are associated with those forces. In parts of the game Link must collect three golden triangles and place them on altars that are arranged identical to the Triforce. Lastly, in the Temple of the Ocean King, the player must draw the Triforce insignia to open a door.
In The Legend of Zelda: A Link Between Worlds, the world of Lorule is introduced. A mirror image of Hyrule in many ways, the land also has its own Sacred Realm and Triforce. The appearance of Lorule's Triforce is inverted, with its tip pointing in a downward direction rather than up. Hyrule's own Triforce is also central to the game's events.
Outside of the Zelda universe, the Triforce has appeared in other various Nintendo games. In Animal Crossing: Wild World and Animal Crossing: City Folk, the Triforce is a collectible piece of furniture. The Triforce symbol also appears in Katrina's fortune teller tent in the original Animal Crossing. In Metroid Prime 3: Corruption, the Triforce is an unlockable sticker that can be placed on Samus's ship.