The Triforce is a shining, sacred, and extremely powerful relic created by the Three Golden Goddesses. It is a central fixture of The Zelda series. As described by Princess Zelda in The Legend of Zelda: The Ocarina of Time:
"The three goddesses hid the Triforce containing the power of the gods somewhere in Hyrule (it would appear that Zelda is mistaken, as the Triforce was actually hidden in the Sacred Realm. This bit of misinformation is probably what lead her to tell Link to unlock the portal to the Sacred Realm, which resulted in allowing Ganondorf to reach the Triforce.). The power to grant the wish of the one who holds the Triforce in his hands.
If someone with a righteous heart makes a wish, it will lead Hyrule to a golden age of prosperity. If someone with an evil mind has his wish granted, the world will be consumed by evil. That is what has been told. So, the ancient Sages built the Temple of Time to protect the Triforce from evil ones.
The Triforce, the sacred triangle, it is a balance that weighs the three forces: Power, Wisdom, and Courage. If the heart of the one who holds the sacred triangle has all three forces in balance, that one will gain the True Force to govern all. But, if that one's heart is not in balance, the Triforce will separate into three parts: Power, Wisdom and Courage. Only one part will remain for the one who touched the Triforce, the part representing the force that one most believes in. If that one seeks the True Force, that one must acquire the two lost parts. Those two parts will be held within others chosen by destiny, who will bear the Triforce mark on the backs of their hands."
In many games Link's mark does not appear until he is an adult. This was most important in Zelda II: The Adventure of Link and Ocarina.
The marks tend to glow when two bearers meet each other, but they also glow at other times, such as when the bearer is experiencing intense moments of Courage, Wisdom or Power, or when it's fundamentally important to the plot.
In Ocarina, Link pulled the Master Sword out of the Pedestal of Time and opened up the portal to the Sacred Realm. Ganondorf entered this portal and attempted to take the Triforce, but it was split into three pieces and he gained the Triforce of Power. The Triforce of Wisdom went to Princess Zelda, and the Triforce of Courage went to Link. When the Triforce is split, the power to grant wishes can be restored to it by reuniting the three pieces, which is what Ganondorf constantly strove to do.
The Triforce (often accompanied by a Phoenix) is also the royal seal of Hyrule. In both Ocarina and The Legend of Zelda: The Wind Waker, Link was able to trigger special events when standing on this royal seal, when he played a three-note song that followed the pattern of the Triforce.
The Triforce has been broken up (and its three parts sometimes further fragmented) many times, and the pieces often scattered all about Hyrule. So far the Triforce has only been wished upon twice in the Zelda series. The first time (chronologically) was by King Daphnes Nohansen Hyrule at the end of Wind Waker, he wished to wash away Ganondorf and the old land of Hyrule to give hope for the future. The second time was by, Link, 'The Hero of Legend' in The Legend of Zelda: A Link to the Past, though his wish was never said out loud, one would assume he wished for the ultimate destruction of the spirit of Ganon and for peace to be restored to the land, as this is what is seen happening after he makes this wish.
Many people don't realize that the Triforce can talk, as he has only done it twice. Once was in the identical opening sequences of The Legend of Zelda: Oracle of Seasons and The Legend of Zelda: Oracle of Ages and the other time was during the ending to A Link to the Past, where it introduced itself as The Essence of the Triforce.
Triforce of Wisdom
One of the three pieces of the Triforce. Usually in the possession of Princess Zelda. It is associated with Nayru, one of the Three Golden Goddesses that created Hyrule. It apparently gives it's user psychic powers and great wisdom.
Triforce of Power
One of the three pieces of the Triforce. Usually in the possession of Ganon. It is associated with Din, one of the Three Golden Goddesses that created Hyrule. It apparently gives it's user great physical strength and magic abilities.
Triforce of Courage
One of the three pieces of the Triforce. Usually in the possession of Link. It is associated with Farore, one of the Three Golden Goddesses that created Hyrule. Though obtaining the Triforce of Courage (or rather, realizing it was already inside Link) gains the player no new abilities, it's believed to be the source of Link's personal courage.
The Legend of Zelda series is one of the most well known and loved video game series ever created. Many fans who have grown up with video games have come to associate the Triforce symbol not only with their love of the series but also with their love of video gaming in general. Subsequently the triforce has become a popular symbol to display on shirts and other paraphernalia that emphasize gaming. Also, the few brave gamers who choose to display their love of gaming in more unconventional ways sometimes choose to get the triforce tattooed somewhere on their body.