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 history 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'.
- In Kara no Kyoukai, 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.
- In Tolkien's Legendarium, Quenya is the ancient language of the first elves, who called themselves the Quendi — "those that speak with voices". The modern elven language, Sindarin, is its descendant.
- 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.
- 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.
- In Far Cry Primal the Wenja, Izila, and Udam all speak variations of Conlang based on what linguists think language would have sounded like roughly 7,000 years prior to the evolution of Proto-Indo European language. PIE would eventually birth the Indo-Iranian, Germanic, Balto-Slavic, and Romance languages that the majority of the modern world now speaks some 10,000 years later.
- In Fate/stay night, certain magi from the Age of Gods are able to 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.