Spiritual Stone

In Hyrule, during the era of Ocarina of Time, Link's quest involves finding the three Spiritual Stones in order to open the Door of Time which has sealed away the Master Sword in the Temple of Time.

The three stones in the game are known as: the Kokiri's Emerald, the "Spiritual Stone of Forest", held by the Great Deku Tree; The Goron's Ruby, the "Spiritual Stone of Fire", a treasure held by Darunia; and the Zora's Sapphire, the "Spiritual Stone of Water", held by Princess Ruto.

During his quest to unlock the Sacred Realm and steal the Triforce, Ganondorf, the evil king of thieves, was refused the Spiritual Stones by each of their respective guardians. In response, he cursed them all with various blights using dark magic. In order to obtain the stones to prevent the Triforce from falling into Ganondorf's hands, Link is required to free the guardians from Ganondorf's malevolence.

"Ye who owns 3 Spiritual Stones stand with the Ocarina of Time and play the Song of Time"

- The Inscription on the Pedestal of Time

Kokiri's Emerald
The Spiritual Stone of Forest, the Kokiri's Emerald, is the first of the three Spiritual Stones Link has to acquire in order to open the Door of Time during his adventures in Ocarina of Time. Ganondorf sends a parasite to retrieve the Kokiri Emerald from the Deku Tree. By relieving the Great Deku Tree of the evil parasite which had infested it, Link proved his courage, leading to the Great Deku Tree entrusting Link with this stone and the fate of Hyrule. However, it was too late for the Great Deku Tree, who dies shortly after giving Link the stone. After Link travels to Hyrule and meets Princess Zelda for the first time, she mentions the Kokiri's Emerald and then sets you on a path to gain the other two stones...

Goron's Ruby
The Spiritual Stone of Fire, more commonly known as the Goron's Ruby, is the second of the three Spiritual Stones that Link has to collect in Ocarina of Time. It is a dearly held treasure of the Goron race. When Link manages to defeat the evil creatures inside Dodongo's Cavern, the Goron's source of food, he is given this stone by Darunia, the "Big Brother" or patriarch of the Gorons. Oddly, even though it is worshiped, some Gorons wish to eat the Ruby, probably because the Gorons love to eat rocks.

Zora's Sapphire
The Zora's Sapphire is the Spiritual Stone of Water which is considered an heirloom of the Zora royalty. This stone was given to Princess Ruto by her mother who said to entrust it to the man she would one day marry. After Link saves Ruto from the insides of the giant whale-like deity Lord Jabu-Jabu, the princess falls in love with the young hero and gives him the Zora's Sapphire as a symbol of their "engagement".

Trivia

 * The Zora's Sapphire is also known as the Zora's Engagement Ring. This is why Ruto gives Link the Zora's Sapphire: as an engagement ring.
 * The emblems of the Kokiri's Emerald, Goron's Ruby, and Zora's Sapphire appear as recurring designs within the Forest Temple, Goron Mines, and Lakebed Temple dungeons of Twilight Princess. Symbols of a similar nature appear within the Arbiter's Grounds, City in the Sky, and Palace of Twilight dungeons, indicating that these are in fact the symbols of these tribes. In addition to this, supporting details can be found in The Wind Waker when Link obtains the Master Sword. If the stained glass windows of the Ocarina of Time sages are observed closely there is a small orb above each sages head. In the center of this orb contains a a symbol different for each sage or race (namely the Kokiri Emerald for Saria, Goron Ruby for Darunia, Zora Sapphire for Ruto, and the eagle on Link's shield in Ocarina of Time for Rauru). This also indicates that the eagle is the symbol of the Hylian race, with the symbol appearing on the Hylian shield, Hyrule Castle, and the Temple of Time, also hinting that Rauru is a member of the Hylian Race.
 * Queen Rutela wears a pendant similar in shape to the Zora Sapphire in Twilight Princess.
 * Using cheats during the game can cause the three stones to move to the center and morph together into one stone when link gets them out in the Temple of Time.
 * In The Wind Waker, the image of the Kokiri Emerald can be found on both the Deku Shield and on Fado's shirt.
 * In The Wind Waker, the image of the Zora's Sapphire can be found in and around areas on Dragonroost Isle.
 * In The Wind Waker, the image of the Goron's ruby appears both the Traveling Gorons clothing and pack.
 * In Four Swords, Four Swords Adventures, and The Minish Cap, The Wind/Fire/Water Elements look respectively similar to at of the Kokiri Emerald, Goron's Ruby, and Zora's Sapphire.
 * In Twilight Princess as well as Ocarina of Time, the image of the Goron's Ruby appears on the skin of the Goron's, whether it be a tattoo they all have, or part of their initial hide.
 * In Ocarina of Time, the image of the Zora's Sapphire appears as a design on the King Zora.
 * In Twilight Princess Queen Rutela also appears to have it around her neck.
 * In Phantom Hourglass and Spirit Tracks, the Adult Gorons have the image of the Goron's Ruby on their upper arms. However, the Child Gorons have the image on their forearms.

Theory
Some have noted the similarities shared by the Pendants of Virtue, the Goddess Pearls, the Spiritual Stones, and the Elements. This could indicate a direct link between these artifacts, or equally, it could simply be an example of the designers' desire to create a feeling of familiarity or a sense of a "shared legend" within the series.