The Common Tongue is the language spoken throughout the Seven Kingdoms, having been introduced by the Andals during their invasion of Westeros six thousand years ago. As the Andals spread across the continent they imposed their language upon their conquests, so that the Andals' language gradually replaced the "Old Tongue" spoken by the First Men. Even in the North, where the First Men had held off the Andal invaders, due to cultural contact and emulation the Northerners gradually adopted the language of their Andal neighbors to the south. Thus for thousands of years, there has been one "common" language throughout all of the Seven Kingdoms. The unification of the Seven Kingdoms in the Targaryen Conquest three hundred years ago served to strengthen this linguistic uniformity even further.
Its full name is "the Common Tongue of the Andals", though it is usually just called "the Common Tongue". Sometimes it is referred to in abbreviated form as just "Common".
Through trade contact (and raiding), many of the wildlings who live Beyond the Wall also know how to speak the Common Tongue. Only those wildlings who live further away from the Wall still speak the Old Tongue of the First Men. Some who live furthest away from the Wall, such as the Thenn people, don't know how to speak the Common Tongue at all, and exclusively speak the Old Tongue.
Over the centuries the Common Tongue of Westeros has spread through much of the rest of the world along trade routes, and thus many people in the Free Cities do know how to speak the Common Tongue, while still primarily using their own local languages. Through trade and shipping, the Common Tongue is even spoken as far east as Qarth. In contrast to the mercantile Free Cities, the Dothraki horse raiders do not engage in trade with Westeros, therefore few of them know the Common Tongue.
In the books
The Common Tongue is virtually the only language spoken throughout the entire continent of Westeros. It is stated that about half of the wildlings can only speak the Old Tongue, generally those that live further away from the Wall or isolated in the mountains. As for the other half of the wildlings, it isn't clear what proportion are bilingual, or perhaps only know the Common Tongue. For that matter, it isn't clear for what proportion the Common Tongue is their primary or secondary language. Nonetheless, the Night's Watch usually doesn't encounter a language barrier with the wildlings that they encounter in their rangings, still relatively close to the Wall.
Behind the scenes
The "Common Tongue" of the Andals is, of course, the English language which book readers and TV viewers observe. Author George R.R. Martin has stated that he is not a linguist like J.R.R. Tolkien was, and thus he never thought out all of the languages in his fantasy world to the extent Tolkien did. Infamously, Tolkien made it a point that characters in his books actually do not speak in English: The Lord of the Rings is presented as if it were a "lost Anglo-Saxon saga" that was rotting in a library until Tolkien "rediscovered it" (much as the actual Anglo-Saxon saga Beowulf was lost for centuries and then rediscovered). Tolkien worked under the conceit that the "Common Speech" (or "Westron") which the major characters in his works speak isn't really English, he just "translated" it into English. For example, "Frodo Baggins" is just a translation of the real Westron name, "Maura Labingi", and "Hobbit" is a translation of the real Westron term, "Kuduk". This extends to the point that certain significant names, puns, or allusions would only fully make sense in "real Westron", and not in English.
Martin has made no such comments about the "Common Tongue" in his fantasy world, and while Tolkien's Middle-earth was intended to indeed be our world in a lost prehistoric age, Martin has stated that the fantasy world of A Song of Ice and Fire is truly an alternate world with no connection to our own. Thus it would seem that the English used in it doesn't "represent" the "real Common Tongue" through universal translation for the sake of the reader (and TV viewer). The Common Tongue just happens to be exactly like English.
Note that, from examples of writing from both the books and the TV series, "the Common Tongue" is specifically American English, and thus follows American English grammar rules and spelling standards (though admittedly, sometimes grammar and word choice can be intentionally archaized to make it sound pseudo-medieval).