Tetraforce

Theory and Hylian Shield
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. Some note that in the final scene that she's portrayed as Shiek in Ocarina of Time, Zelda did say something along the lines of getting the "True Force" if he or she had equal Power, Wisdom and Courage. The "True Force to Govern All", however, is used time and time again when referring to the complete Triforce, which consists of just three pieces. In later redesigns of the Hylian Shield (Like the one in Twilight Princess or the Hero's Shield in Majora's Mask) the bird and the possible fourth piece is missing.

Naming
Possibly though, "Tetraforce" is not a correct name for the holy triangle if there was a fourth piece. In the very first The Legend of Zelda, there were only two pieces of the Triforce, the third one not being introduced until The Adventure of Link. It is unlikely that Nintendo was already planning the story of The Adventure of Link before they knew how the first game would do, especially since it came out in a time when the video game industry was still recovering from its near destruction a few years earlier. If so, "Triforce" refers to the triangle-shape and not the number of pieces of the (since The Adventure of Link) whole thing, in which case the Triforce stays the Triforce no matter the number of pieces.

A Fourth Goddess
The known legend in the history is that the Triforce was created in the beginning by three goddesses which leaves no presence of a fourth goddess that created the supposed fourth piece. Note that there are more gods mentioned throughout the series though, but they are never connected to the three who created the Triforce.

Goddess of Time
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 Princess Zelda's mention of a Goddess of Time in Majora's Mask as evidence. However, detractors point out that there's also some evidence to suggest that Nayru and a goddess of time are the same being. The Goddess of Time may be Farore, however, as Ciela states that she is the spirit of courage and of time, though no mention of the three goddesses is made in Phantom Hourglass.

Sand Goddess
Other theorists claim the "Sand 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 supposedly Evil Goddess may have created life that would break the law and as a result she could have been banished along with her piece of the Triforce.

Zelda
Some people believe that Princess Zelda is the 4th goddess, and represents the fourth piece of the Tetraforce, hence her pirate reincarnation name, Tetra (from The Wind Waker) which is Greek for the word "four" (although this most likely has to do with the fish species tetra and not the Tetraforce). There is a symbolic cut scene from 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.

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 Princess 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.

Kindness
The fourth piece could represent "Kindness", as Majora's Mask featured four Great Fairies in the four directions, called the Great Fairy of Courage, the Great Fairy of Wisdom and the Great Fairy of Power. The fourth one though was called the the Great Fairy of Kindness. However, these Great Fairies represent the qualities of the Four Giants, so it is unknown if there actually is a verified connection.

Light Spirits
In Twilight Princess there are four Light Spirits (Eldin, Faron, Lanayru and Ordona) each representing their own province and three of them represent a Goddess Eldin represents Din, Lanayru represents Nayru and Faron represents Farore. The fourth spirit, Ordona also represents Farore. When talking to Lanayru, it says "We three spirits were ordered by the Goddesses to seal away the evil power" Which is implying that either The creators have made it clear that Ordona is supposed to represent the second half of Farore's name
 * Ordona was not present at the time of the incident.
 * Ordon and Ordona are not considered as part of Hyrule, as mentioned later in the game by Shad, who says "I hear he's not from Hyrule proper at all".

Shadow
The most little thought fourth goddess may be Twinrova as she raised ganon in the OoT, but unlikely because she doesn't have any true god powers.

Fierce Diety
It is believed that when one has the complete Tetraforce, it is impossible to control its power. People use Fierce Deity Link from 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.

Triforce of Termina
Skull Kid is shortly seen above what seems to be the Terminian version of the Triforce. It is possible that this may have something to do with the Tetraforce. The Stone Tower's carvings make this very likely.

Light Force
Some timeline theories may argue that the Light Force from The Minish Cap, is the completed Tetraforce. This is possible if The Minish Cap takes place before all other Zelda games, and some unknown event after The Minish Cap and before Ocarina of Time causes the middle piece of the Tetraforce to unbind itself and become lost. The Triforce could have earned its name, representing an incomplete Light Force, or otherwise Tetraforce. Another possible theory is that at the end of The Minish Cap, Vaati states that when he tried to absorb the Light Force from Princess Zelda, he could not absorb all of it. This could mean that the portion of Light Force he did not obtain became the "normal" Triforce. This portion of the theory does not hold very well, considering the properties of the Light Force do not match those of the Trifoce and the many Triforce crests scattered throughout Hyrule in the game. However, it could be argued that the properties of the Triforce could change once the missing fourth piece is united with it.

Triforce of Shadow
There is a theory that the dark space in the middle of the triforce is a supposed "Triforce of Shadow". Though there is not much evidence that there is a Triforce (or tetraforce) of shadow. It also helps prove Dark Link's creation.

The Great Force
According to unused text in the game Mario & Luigi: Superstar Saga, Link would make a cameo at Starbeans Café the first time the player selected to drink the Teeheespresso. The Triforce would also, somehow, be collected in the game and could be given to Link in exhange for the Greatforce ho was likely the fourth piece OR the completed Triforce.. This was, however, changed, presumably to avoid confusion as they has declined it on several occasions (see below).

Officially
Despite all this, Nintendo and Shigeru have vehemently denied it on multiple occasions. The fourth Triangle was removed from later redesigns of the Hylian Shield and also from the Hero's Shield from The Wind Waker and Majora's Mask. Officially, the Tetraforce does not exist and it should be considered fan speculation.