It seems reasonable that if you trace back the history of humanity far enough, you'll eventually find the first group of humans. Presumably they would have spoken a specific language among themselves, and this would have become the basis for all human language thereafter. Some writers take this notion and expand on it, giving this ur-language or its speakers special properties.
The Primordial Tongue is likely to be a Lost Language with strange features, some of which may bequeath supernatural powers. Because of its status, it may be a Language of Magic or of Truth. Its speakers will likely be Omniglots, able to parse any language they hear by comparing it to the original. Even if it has no special abilities, learning it will be of great value to scholars.
In the real world, there's considerable debate over whether a primordial language could exist. The early prehistory of Homo sapiens is complex and full of holes, and interacts with numerous other hominids, now extinct, who might have influenced our species in a number of ways. Even if one original language did exist, what we know of the time-scale and languages in the world today indicate that we would probably not be able to smoothly translate between them in the way that is shown in fiction.
In a setting where humanity is not collectively very old (especially if they were explicitly made by gods), the Primordial Tongue may also be a Classical Tongue, and may be the ancestor of the Common Tongue.
Related is the Curse of Babel, where God or an equivalent puts an end to the primordial tongue and make humanity speak different languages.
- Atlantis: The Lost Empire: Atlantian is presented as this, and demonstrated when Kida is able to learn several modern languages by hearing snippets of them and reverse-engineering them. Milo, likewise, is able to learn Atlantian by reversing the same process.
- The Fifth Element: The apparent gibberish Leeloo speaks, which Cornelius insists is the divine language that was once spoken throughout the universe, indicating it's a cosmic version of this trope. Using it, Leeloo is able to figure out how to speak English after about a day of exposure, and notably is able to read a sign early on, despite having no way of knowing what the letters would mean.
- The Librarian: Quest for the Spear features a book written in "The Language of the Birds" which is stated to be the language spoken before the Tower of Babel.
- The idea that there was once an "Adamic language", which is what Adam and Eve would have spoken in Eden and what people kept speaking until the fall of Babel, was a very popular belief in medieval Europe. Attempting to reconstruct it was a fairly long-running pursuit in mysticism, since it was believed to have mystic powers, and some people believed that children raised without exposure to modern languages would start speaking it as well. Others believed that Adam and Eve would have spoken Hebrew, however.
- Artemis Fowl: The fairies' language, Gnommish, is described as the ancestral language to every tongue on Earth — including American Dog, somehow. This is the in-universe reason behind the fairies' ability to speak all languages, including animals'.
- The Garden of Sinners: The Unified Language is the language of ancient Babylonia, back when no other languages existed and humans, animals, and non-living things all spoke. It is able to commune directly with the soul, allowing it to function as a form of telepathy or hypnotism that can force people to do or see things. In the modern day, the only person able to speak it is Satsuki Kurogiri after he was kidnapped and tampered with by faeries.
- Space Trilogy: The language of the Hrossa of Mars (Hressa-Hlab in Out of the Silent Planet) is said to be "Old Solar" (Hlab-Eribol-ef-Cordi, "a common speech for all rational creatures inhabiting the planets of our system") in the sequel Perelandra, allowing Ransom to speak to the natives on Venus. In the third novel, That Hideous Strength, it's revealed to be the first language ever created. However, although once spoken on Earth (Thulcandra) as well, it was lost when the Earth's Guardian Angel (or Eldil) became the Devil, and "no human language now known in the world is descended from it".
- Tolkien's Legendarium: Primitive Quendian is the ancient language of the first elves, who called themselves the Quendi — "those that speak with voices". The modern elven languages, Quenya, Sindarin, Telerin, Sylvan, etc, are its descendants. Since Tolkien was a professional linguist, he got this process right — the Noldor believed Quenya was closest in features to Primitive Quendian, but while Quenya and Telerin are closely related (due to the geographic proximity of their speakers), Sindarin and Sylvan have diverged significantly, and Tolkien gave a few examples of words from other Elven tongues that show regular divergence from what would become Quenya. (Needless to say, while Quenya and Telerin may be closest to Primitive Quendian, they are not mutually intelligible.)
- Nova: "First Horse Warriors" discusses the ancient Yamnaya people, believed to have spread out of what is now southern Russia behind a black plague pandemic in about 3000 BCE. They are theorized to have been the original speakers of Proto-Indo-European, with the plague explaining what happened to the other language groups of the areas they spread to.
- Ars Magica has the lost language used by the Biblical Adam to name all the creatures of Earth after their creation. It can only be rediscovered through painstaking forensic linguistics or by finding the Garden of Eden or Cain, but would enable a speaker to shape destinies, empower their own magic, or forge permanent Sympathetic Magic links by bestowing Names.
- Warhammer Fantasy has the long-lost language spoken by the Old Ones who created the sapient species of the world. Its closest descendant is the Language of Magic of the High Elves, but fragments also survive in the Dwarfs' Runic Magic and in the Black Speech of Daemons.
- The Elder Scrolls series has Ehlnofex, the language of the Ehlnofey, Precursors of both Mer (Elves) and Men, who were themselves immediate descendants of the world-creating Et'Ada. It is very much a Language of Magic and is always written in entirely capital letters. Perhaps the most famous word is CHIM, the in-universe concept of Ascending to a Higher Plane of Existence where one becomes aware of the nature of Anu's Dream, but exists as one with it and maintains a sense of individuality. It would eventually evolve into Aldmeris, the Classical Tongue of the setting, which itself divided into numerous other languages, including the Common Tongue Cyrodiilic/Tamriellic, as well as multiple Lost Languages including Dwemeris and Falmeris.
- Escape from Monkey Island: The central MacGuffin is the Ultimate Insult, a magical device that so thoroughly and ruthlessly mocks a subject that they permanently become an Extreme Doormat afterward. In the last act it's revealed to work by speaking in a primal tongue that bypasses the limitations of normal language and speaks directly to the soul: specifically, it's monkey jabber.
- Far Cry Primal: The Wenja, Izila, and Udam all speak variations of a Conlang based on what linguists think the distant ancestor of Proto-Indo European (PIE) might have sounded like circa 10,000 BCE. PIE would eventually birth (among others) the Indo-Iranian, Germanic, Balto-Slavic, and Romance language families that much of the modern world now speaks.
- Fate/stay night: Certain magi from the Age of Gods can speak "Divine Words", allowing them to cast powerful spells with a single word or gesture when it would take entire teams of modern magi in the present day chanting for several minutes together to produce the same effect. Caster abuses this with abandon to bombard her foes with Beam Spam, vaporizing anyone unwary of her.
- Castlevania (2017): Adamic, the language that Adam and Eve spoke in Eden and from which all modern languages derive, turns up in "The Last Spell" when one of the books the heroes are investigating is revealed to be written in a form of it, which Sypha needs additional time and resources to translate.