The Divine Beasts are mechanical constructs from The Legend of Zelda: Breath of the Wild. There are four Divine Beasts; Vah Ruta, Vah Medoh, Vah Rudania and Vah Naboris, each of which were corrupted by Calamity Ganon during the Great Calamity causing disasters all over Hyrule. Link must free them from corruption by entering their innards and returning them to their owners, the four Champions. The Beasts' innards function as dungeons and Link has to battle their bosses. Once set free, the four Divine Beasts will help Link confront and defeat Calamity Ganon.
in the Tabantha Frontier
in the Gerudo Desert
on Death Mountain
in the East Reservoir Lake
The Divine Beasts' innards are more expansive than the shrines, and different than traditional dungeons of the The Legend of Zelda series. They are shorter but have wide chambers. All the Divine Beasts have a set of terminals, which Link has to activate. Once done, he is then able to turn on the Main Control Unit leading to the boss fight of each Divine Beast, respectively Windblight Ganon in Vah Medoh, Thunderblight Ganon in Vah Naboris, Waterblight Ganon in Vah Ruta and Fireblight Ganon in Vah Rudania.
Each Divine Beast has its own unique controllable function, directly correlated to its form and design. The functions can be controlled while inside each beast's innards by downloading the map on the Sheikah Slate. After freeing each beast, Link may not enter and therefore may not adjust the beasts after leaving them. Vah Medoh, a mechanical eagle, has the ability to have its roll adjusted which puts the dungeon at a tilt. Vah Naboris, a mechanical camel, has individually controllable rings in its stomach that each contain a line. When properly adjusted, the lines are charged with a high electric current. Vah Rudania, a mechanical salamander, can completely turn its body at a 90° angle. Vah Ruta, a mechanical elephant, can move its trunk to spray water in different areas. All four beasts make unique calls, with Ruta's and Medoh's bearing the signature sounds of their respective animals. Once Link successfully liberates a Divine Beast and the Champion's spirit, he can no longer reenter the respective beast, as it would rest on a certain location, taking aim at Hyrule Castle, which Link would be warned that it is dangerous to approach before turning away.
During the quest "EX Champions' Ballad" unlock upon installing "The Champions' Ballad" DLC pack, a fifth Divine Beast at the Shrine of Resurrection at the Great Plateau, taking the name "Final Trial". Not much is known about this and who it is intended for, although it holds the location of the Master Cycle Zero with the Master Cycle Zero taking the form of a horse.
According to an ancient legend of the battle with Calamity Ganon ten thousand years before Breath of the Wild, there was a period of time when the Sheikah developed a technologically advanced civilization in Hyrule. Knowing Ganon, the ancient evil and old enemy of the Royal Family of Hyrule, would return, the Sheikah developed a plan to confront him with their best technology. This lead to the creation of the four powerful machines that became known collectively as the Divine Beasts and an army of autonomous weapons called Guardians all designed to protect and aid the current Princess of Hyrule and Hero Chosen by the Master Sword in confronting and sealing Ganon. Each Divine Beast was given to one of four races: Vah Ruta to the Zora, Vah Naboris to the Gerudo, Vah Rudania to the Gorons, and Vah Medoh to the Rito. Each race selected a Champion to pilot their Divine Beast. With their plan in motion, a great battle ensued with the Guardians and Divine Beasts supporting the Hero, who defeated Ganon, allowing the Princess to seal him once more. The Sheikah technology was hailed for a time as having power akin to the gods. However, people eventually began to fear such technology being misused. This caused the Sheikah to become marginalized in Hylian society for a time, leading to a schism among the Sheikah over their treatment. Most Sheikah chose to heed the people's fears and buried their technology, living simple lives. However, others angered by said mistreatment despite the Sheikah doing so for Hyrule's benefit broke off to form the Yiga Clan, who swore loyalty to Ganon. As a result, much knowledge of the Ancient technology was lost and faded into legend.
A century before Breath of the Wild, it was foretold by the fortune teller that the return of Calamity Ganon was inevitable. This sparked a search for the fabled ancient Sheikah relics, including the Guardians and the four Divine Beasts. Many believed them to be simply a legend, but their discovery made this legend a reality for them. Realizing this, King Rhoam and Princess Zelda formed a new group of Champions inspired by the unnamed Champions of legend. Commanded by Princess Zelda, four elite warriors were selected from across the kingdom and were tasked with the duty of piloting the Divine Beasts to assist Link, Princess Zelda's appointed Knight and the Hylian Champion selected by the Master Sword. Calamity Ganon reappeared and attacked Hyrule Castle, and having learned from his previous defeat, he sought to turn Hyrule's ancient technology against it. He took control of the Guardians and the Divine Beasts, killing four of the Champions and King Rhoam in the process. Link was injured protecting Zelda from Guardians and placed in medical stasis within an ancient Sheikah medical facility Zelda's research team had previously uncovered. Guided by the voice of the spirit of the Master Sword, Zelda entrusted the blade to the Great Deku Tree before sealing herself and Ganon in Hyrule Castle. This event became known to Hyrule's people as the Great Calamity.
A hundred years later, however, the Divine Beasts reactivated and began causing trouble for the races they were entrusted to. Vah Ruta threatened to flood Zora's Domain with torrential rain. Vah Rudania rained down magma bombs forcing the Gorons to abandon their mines on Death Mountain. Vah Naboris wandered through Gerudo Desert kicking up sandstorms and attacking any that drew close with lightning. Vah Medoh appeared in the skies above Rito Village attacking any Rito who flew too high. They were ultimately disabled and purged of their corruption by Link.
After freeing Vah Naboris, the Gerudo Champion Urbosa explicitly states that Naboris was named after Nabooru. Additionally it is also stated in a Zora Stone Monument authored by Zora King Dorephan that Vah Ruta is named after Princess Ruto who's history as Sage of Water and Lord Jabu-Jabu's attendant is also referenced. Vah Medoh stands out, as it is the only Divine Beast to not be named after a sage from Ocarina of Time, but a sage from The Wind Waker as the Rito does not exist until the latter game.