Destiny Stones

Destiny Stones are like the "cores" of the Digital World, meaning that if they are tampered with, the Digital World suffers. The destruction of all the Destiny Stones would mean the consequential destruction of the Digital World; the destruction of one begins the "warping" of the Digital World, causing Digimon to appear in the Real World. Destiny Stones are first mentioned in the whole series in episode 32 of Digimon Adventure 02, If I only had a Heart.

Digimon Adventure 02
Destroying the Destiny Stones is originally one of many schemes by Mummymon and Arukenimon to destroy the Digital World. In "If I only had a Heart" Mummymon and Arukenimon first come across a Destiny Stone and begin attempting to destroy it. They are delayed by the DigiDestined, who attempt to stop them from further damaging the Digital World. During this, BlackWarGreymon is having a discussion with Agumon about life (see episode: "If I only had a Heart"). However, the discussion is cut short when several blows are made to the Destiny Stone, causing BlackWarGreymon--for unknown reasons--a great amount of pain. BlackWarGreymon flies away from Agumon and destroys the Destiny Stone, despite the attempts to stop him by the DigiDestined.

BlackWarGreymon proceeds to destroy all the Destiny Stones, to the delight of Mummymon and Arkenimon. The DigiDestined try and fail to stop him every time. Initially BlackWarGreymon's motive to destroy them is the pain he is caused when they are tampered with; in Destiny in Doubt a new motive is discovered. In Episode #34, when BlackWarGreymon destroys a Destiny Stone, a (pained) Digimon appears in the sky briefly. BlackWarGreymon--in beleif that his destiny is to only destroy, and become the strongest Digimon, the conversation with Agumon forgotten--believes this Digimon to be his only worthy opponent. From then on BlackWarGreymon destroys the Destiny Stones to ultimately battle this Digimon.

Digimon Sovereigns
Azulongmon, the Digimon seen in the sky when a Destiny Stone is destroyed, is the Digimon Sovereign of the Eastern Hemisphere.