As the most powerful Pokémon in the game, possessing stats roughly equivalent to Arceus, Asphere is very capable in every field. In its Vega Forme, it has balanced Attack, Defense and Special Defense, with its Special Attack and Speed being as high as those of its other Formes. Coupled with its diverse movepool and powerful signature move, Nova Inferno, the only other Pokémon in the game with comparable special attacking prowess is Mewtwo.
Asphere seems to represent various stars and stellar atmospheres - the word 'atmosphere' being the most likely basis of its etymology. Its Altair Forme, possessing the power of the constellation Aquila, has a Dragon/Ice typing. It is named after Altair, the brightest star in the Aquila constellation. Its Sirius Forme is a Dragon/Electric-type, and is likewise named after the brightest star of Canis Major. Finally, its Vega Forme is named after the brightest star of Lyra.
In all games in which it appears, Asphere resides at the lowest floor of the Sphere Ruins, where it can be battled at Lv90. Interestingly, it is capable of human speech, and shows both surprise and respect for the player being able to reach it. It challenges the player, claiming to fight with the full power of its correspondant constellation.
In Pokémon Altair & Sirius, Asphere's true identity is suggested to be that of a guardian deity, or possibly a ruler, of an old civilation that lived in Sphere City. This is shown through Braille writings found within the Sphere Ruins.
In Pokémon Vega, while the real Asphere is still waiting within the (now displaced) Sphere Ruins, Turner uses two cloned Asphere in their Altair and Sirius Formes, among several other unusual Pokémon made using Mewtwo's technology. After Turner is defeated, he flees, leaving both clones to rampage out of control. The rival proceeds to calm them down with his Capture Styler before setting them free. What becomes of the clones after these events is unknown.