The Sages of the Master Sword are sages of Hyrule in The Legend of Zelda: The Wind Waker. They pray to the gods to keep the power to repel evil inside the Master Sword, the blade of evil's bane used to destroy the king of evil, Ganondorf, if he ever were to escape from the Sacred Realm. They pray to the gods by playing the harp and the violin inside the Earth Temple and Wind Temple respectively. The Earth Sage keeps the sword's edges in shape and the Wind Sage keeps the sword's gem glowing and the blade sparkling with evil-destroying power. Eventually, Ganondorf broke the seal and attacked Hyrule, but a hero did not appear to stop the malevolent warlock. The gods flooded Hyrule in response to seal away the evil.
Hyrule became the Great Sea and the survivors settled on the mountaintops of Hyrule, now islands, while Ganondorf, his army, and the rest of Hyrule were sealed in time on the ocean floor. Ganondorf managed to escape, and sent Molgera to kill Fado, the Sage of Wind, and Jalhalla to kill Laruto, the Sage of Earth. The two Sages were assassinated and the Master Sword fell powerless. The power vanished, the gem went black and the edges went dull. However, their magical instruments survived and were passed down their family lines. After Link from The Wind Waker took action, their spirits appeared before him and instructed him to awaken their descendants as the new sages. Makar the Korok (Fado's successor) and Medli the Rito (Laruto's successor) were awakened and restored the power to repel evil to the Master Sword. Medli restored the edges and Makar restored the power and gem.
Scrapped Third Sage
It is possible there was a third scrapped sage for one of the missing dungeons. It is strange that there are only two sages of Earth and Wind which feels like its missing a third link because Earth feels out of place next to Wind, and Nayru's Pearl has no dungeon attached to it leading to this idea. The third sage could of been Sea, Sky or Ice. There was an Ice, Wind and Sky dungeon to be put in Ocarina of Time, but were ultimately scrapped however Wind was implemented into Wind Waker. The sage some theorize it to be was actually Aryll since she plays an important role in the game, goes missing after her rescue for the entire rest of the game until the end which seems odd and like Saria, plays an intimate role in Link's life. It would be fitting if Aryll turned out to be a third sage since she is Hylian as well, representing another race of Hyrule and thus would have her fate intertwined with her brother's role as chosen hero supporting him by being a Sage that prays to give power to the Master Sword.
The sages being of Earth, Wind and Sea would fit the sea travel theme of the Wind Waker as well.
Another thing to point out that since the Earth Temple and Wind Temple representing worship areas for Farore and Din seems heavily implied, a Sea Temple for Nayru would be plausible. Aryll being the Sage of the Sea would fit with her strange affinity with Seagulls and the seafaring theme of The Wind Waker.
Master Sword Monks
In The Legend of Zelda: Breath of the Wild, there are seven unnamed Sheikah Monks that oversee the "Trial of the Sword" which unlocks the power of the Master Sword depending on how many floors have been completed unlocking its true power once all floors are completed. It is possible the Master Sword Monks are the current incarnation of the Sages of the Master Sword presumably praying to keep the power to repel evil within the Master Sword. Presumably the number of Sword Monks is due to Hylia's influence as the Sheikah Monks are devoted followers of Hylia with the Sword Monks being chosen by her to oversee the important trial and give their prays to the gods to empower the the Master Sword with the spark to repel evil.