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 similar offensive prowess is Mewtwo. Due to its extremely wide movepool and access to both Swords Dance and Nasty Plot, among many other useful moves, it is a very versatile Pokémon that allows for many different playstyles, and a worthy reward for reaching the end of the Sphere Ruins.
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.
While Asphere itself is encountered at Lv90, the wild Pokémon in the Sphere Ruins' lower floors can be encountered at even higher levels, up to and including Lv100.
Asphere's other signature move, Icebolt Wave, can only be learned by it in Altair/Sirius, as it is the signature move of those Formes. As such, Turner's clones do know the move.
Asphere's stat total of 720 is the highest of all the games it's been in, equal to that of Arceus.
The Asphere clones used by Turner in the final battle have base stats far higher than what's reasonable for an obtainable Pokémon. This may have been intended to simulate EV-trained versions of them.
Asphere is one of a select few Pokémon shown to be capable of human speech in the game. The others are Mewtwo and Aldina.
Turner's clones do not seem to be able to speak, only letting out primal roars during their rampage at the end of the game.
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 Altair/Sirius, Gordon will mistakenly call Asphere "Asmodeus", which is actually the name of a mythological demon king. This is likely only a gag without any actual relevance.
In Vega's Mirage Battle System, Sylvia uses a Vega Forme Asphere.