If you look at the OoT Hylian shield, there is an upside down triangle under the red bird; some people think that that fills in the hole in the middle of the triforce (four parts, thus, tetraforce).

Also, 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's Mask. Also, if one looks at the Fierce Deity's armor, there is a triangle on the armor over his left pectoral muscle, this could possibly represent a fourth piece, or perhaps it shows the triforce compleated with a fourth triangle for the center hole.

Some timeline theories 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.

These are the theories for the fourth piece's fourth goddess:

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 (Hope)

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 was called the the Great Fairy of Kindness.

This is from research on various sites.