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Abyssal: The tongue of chaotic evil Outsiders such as Demons
Ancient: The tongue that predated Common
Animal: The "language" of common animals
Aquan: The tongue of water-based beings such as Water Elementals
Auran: The tongue of air-based beings such as Air Elementals
Celestial: The tongue of good-aligned Outsiders such as Angels
Common: The most common tongue of the Prime Material Plane
Darkspeak: The tongue of most Underdark inhabitants; also known as Undercommon
Draconic: The primary tongue of Dragons, Kobolds, Troglodytes, and Lizardfolk
Dwarven: The tongue of all Dwarves
Elven: The tongue of all Elves
Giant: The tongue of Giants, Minotaurs, Trolls, and Ogres
Gith: The tongue of the Githyanki and Githzerai
Gnoll: The tongue of Gnolls and Flinds
Gnome: The tongue of all Gnomes
Goblin: The tongue of the goblinoid races such as Bugbears, Goblins, 
        and Hobgoblins
Halfling: The tongue of all Halflings
Ignan: The tongue of fire-based beings such as Fire Elementals
Infernal: The tongue of lawful evil Outsiders such as Devils
Insectoid: The tongue of insects
Magical: The tongue used as the verbal component of spells
Orcish: The tongue of all Orcs
Sylvan: The tongue of Fey creatures such as Pixies, Satyrs, Centaurs, Dryads,
        Leprechauns, and Bronwnies
Terran: The tongue of earth-based beings such as Earth Elementals
Thieves Cant: A verbal and gestural language generally known only to Rogues
* There are certainly exceptions to the generalizations made in the table above
  about who speaks what language.