Tetraforce
'Tetraforce' is a name given to the Triforce by people that believe that the Triforce has four pieces instead of three. Belief in the Tetraforce's existence was sparked by people who noticed on Link's Hylian Shield the pedestal on which the bird is standing on as a fourth triangle and jumped to the conclusion that it was a fourth piece of the Triforce that had been separated from the others, hence the name 'Tetraforce'. Some say the bird on the shield is carrying the 4th piece to the rest of the Triforce. The known legend in the history is that the Triforce was created in the beginning of time by 3 goddesses which leaves no presence of a fourth entity that created this fourth piece.
This theory is widely regarded as being foolish, though there are still some people in the community that support it. It is speculated by some that the missing "piece" represents time, and some supporters have used Zelda's mention of a Goddess of Time in Majora's Mask as evidence. However, detractors point out that it is only mentioned in passing while she bids goodbye to Link; there's also some evidence to suggest that Nayru and a goddess of time are the same being.
Other theorists claim the 'Goddess' represented by the Desert Colossus could be viewed as evidence for a fourth piece. This Goddess would explain the existence of the Gerudo, as while Farore created life to uphold Nayru's law, this Evil Goddess may have created life that would break the law.
Another possibility is that the fourth inner piece represents hope; this theory comes from the oracle games. Since there is the oracles of seasons, ages, and secrets each corresponding to a piece of the Triforce (seasons=power,ages=wisdom, and secrets=courage) and eventually Zelda comes along claiming to be the oracle of hope, possibly meaning that she is corresponding to the fourth piece of the Tetraforce. It is also possible that Zelda had actually always been a goddess of hope sent to Hyrule to be continuously reborn as the Princess of Hyrule.
Some people believe Zelda is the 4th goddess, and represents the fourth piece of the Tetraforce, hence her pirate reincarnation name, Tetra (from The Legend of Zelda: The Wind Waker)(although this most likely has to do with the fish species Tetra and not the tetraforce.). There is a symbolic cut scene from The Legend of Zelda: The Wind Waker, when Tetra's necklace piece unites with a small triangle given to her from the King. This could be seen as a visual metaphor of Zelda owning the fourth piece of the Tetraforce and having the ability to unite it. Unfortunately for these believers, Link also came across the instance of needing to combine his Triforce of Courage thus proving that it is unnecessary to possess the fabled 4th piece to combine the shard of a piece of the Triforce.
It is believed that when one has the complete Tetraforce, it is impossible to control its power. People use Fierce Deity Link from The Legend of Zelda: Majora's Mask as an example. It is believed that in a past incarnation in the Zelda timeline, Link had found the complete Tetraforce and became immortal. He could not control his power, and he caused much destruction, so the Goddesses created the Song of Healing, which was used to seal the Fierce Deity inside a mask. It is thought that Fierce Deity Link may have been evil in the past since the child wearing Majora's Mask on the Moon tells you that you will be playing the bad guy when he hands you the Fierce Deity Mask.
Some timeline theories may argue that the Light Force from The Legend of Zelda: The Minish Cap, is the completed Tetraforce. This is possible if The Legend of Zelda: The Minish Cap takes place before all other Zelda games, and some unknown event after The Legend of Zelda: The Minish Cap and before The Legend of Zelda: Ocarina of Time causes the middle piece of the Tetraforce to unbind itself and become lost. Before this point, no one knew the Light Force was made up of four pieces - it only appeared to be one piece. The Triforce could have earned its name, representing an incomplete Light Force, or otherwise Tetraforce. This portion of the theory does not hold very well, considering the properties of the Light Force do not match those of the Trifoce. However, it could be argued that the properties of the Triforce could change once the missing fourth piece is united with it.