From The Legend of Zelda and its sequels, the Triforce was a set of "golden triangles possessing mystical powers" supposedly made by three gods from a distant nebula. In the Golden Land the Triforce has the power to grant a mortal's wish.
Generally there are three aspects of Triforce, which unite to make a whole:
- Wisdom: "The Keeper of Knowledge"; associated with Princess Zelda.
- Power: "The Forger of Strength"; usually associated with Ganon.
- Courage: "The Juror of Courage"; associated with Link.
In Zelda I, Ganon stole the Triforce of Power, and Zelda, afraid he might take the Wisdom one as well, divided it into eight pieces and hid them all over Hyrule.
Zelda II is about Link's quest to retrieve the Triforce of Courage to reunite the Triforce and revive Princess Zelda.
In Zelda III, the quest is to retrieve the Triforce from Ganon who has used its power to wish for the Golden Land itself, making it the Dark World.
I don't think there was Triforce in the Gameboy game Link's Awakening, and I haven't played Ocarina of Time yet...
Update! SpudTater informs me:
There is indeed a Triforce in the Ocarina of Time. As I recall, Ganondorf tricks Link into opening the Sacred Realm. Then G enters the SR, and tries to take the TF. However, he only gets Power. The other two pieces go to Zelda & Link.
G wasn't able to get the other pieces because his character lacked these two aspects. So he captures Zelda, and uses her to lure Link into a confrontation, planning to capture them both and forcing them to wield their TF pieces for his purposes.
Milen confirms this, and adds:
Link's Awakening does not focus on the Triforce (though a Triforce piece makes a cameo as the "Piece of Power" special item that temporarily doubles attack strength).
The Triforce plays no part whatsoever in Majora's Mask, by the way.