For fantasy writers and D&D dungeon masters alike having a "common tongue" language is a bit of lazy storytelling just so the person telling the story doesn't have to worry about different languages regularly.
However you may have noticed that when reading books by professional writers they put more thought into how they present languages, possibly even having characters speaking in different languages in order to relay secrets.
For example Tolkien used a common tongue in his works, but he also created multiple languages to fulfill his needs. He also understood that languages evolve and change over time. He created 15 different dialects of elvish, as well as languages for the orcs, ents, hobbits, and more, and his creation of the languages influenced place names, the names of characters, and how he presented languages.
In the real world English is currently the equivalent of the common tongue on the global scale. Depending upon what part of the world a person lived in, and at what point in history, the common tongue would have been something different however. Eg. French, German, or Latin at varying points would have been considered a common tongue in Europe. In the future it is conceivable that English will fade in popularity and a language like Chinese (specifically Putonghua, aka Mandarin) will become popular in the future.
However as you've probably guessed not everyone in Europe spoke Latin, despite the Romans doing their utmost to conquer everyone. If someone was educated or a merchant there was a good chance they also spoke Latin, but the craftsmen or the peasantry would barely speak a word of the language beyond ita vero et minime (yes and no).
Granted, if you're writing fantasy then some readers won't particularly care if you skip such realisms, but if you're writing historical fiction then you're going to want to be as accurate as possible.
It really depends on how much realism you want in your story.
Here's a few questions to think about to improve realism:
1. What is the common tongue called?
Eg. In Korovia then Korovian is the common tongue in that part of my world. But if a person travels to Al-Kazar or the Azagolian Empire then Al-Kazarian or Azageul is the common tongue in those regions.
2. Are there regional languages or older languages which are spoken more often in certain areas?
Eg. In Korovia the region is Slavic themed, but some people still speak Middle Korovian, Old Korovian or even Ancient Korovian. Thus I like to borrow bits from languages like Romanian and Russian.
3. Are the towns or villages based on words from the language, or are they based on words from a different language?
Eg. When naming a number of the towns and villages of Korovia I opened a map of Romania many years ago and then I browsed the names of villages, creating new names which sounded similar to existing places.
4. Does magic use a specific language?
Eg. In Korovia, yes. This is where I like to borrow words from Esperanto. Esperanto is a made up language created by a Polish academic in 1887. It is mostly spoken by language scholars, but for my purposes it makes for a good "language of magic".
Note - J. K. Rowling did something similar for magic in her Harry Potter series. She used Latin for all of the magical words.
5. Do you have a map showing where different languages are spoken?
Something to consider doing. Very useful.
6. Are there any hybrid languages?
Hybrid languages are usually spoken near the borders between major countries, but they can also happen when two cultures are mixed over a long period.
Eg. Yiddish is a hybrid of German and Hebrew. Similarly Alsatian is a language spoken mostly in eastern France that is a mixture of French, German and even Yiddish. Likewise in modern times Denglisch is a mixture of Deutsch and English, the result of so many English speakers living in Germany these days.
Note - If you watch the film Waterworld you will note that the characters refer to one language they encounter as Portugreek, which is apparently a hybrid of Greek and Portuguese.
7. What are the languages actually called in their native tongue?
Eg.
Germans don't actually speak "German", they speak Deutsch. Russians
speak Russkiy. The Chinese actually speak over 300 languages native to
China, Putonghua (Mandarin) is just the official state language, but
depending upon the region a person is in there are different dominant
languages.
8. Does each language have a traditional greeting or way of saying goodbye?
Useful to use these.
Eg.
In Forgotten Realms the common greeting is "Well met", but that
wouldn't necessarily be the only way of saying hello and other greetings
might be more popular in specific regions.
9. Is there more than one common tongue in your world?
When cultures collide it might make sense to just call the two languages by their names, as opposed to calling both of them "common".
10. Do you have a separate alphabet for one or more languages?
Below... the Dwarven alphabet created by Tolkien.
Final Thoughts
Not all writers (or Dungeon Masters for playing Dungeons & Dragons) are into this level of realism within their writing, but for those that are the extra details and realism helps to immerse the reader (and players) within the fantasy world. Or in the case of historical fiction, within a time and place.
For example if you're writing a book set in the Holy Roman Empire the people there don't speak English obviously, and they don't speak Latin either. They're speaking German (Deutsch). Thus they wouldn't use words like "Hello!" they would say things like "Good day!" instead because "Guten Tag!" is the standard greeting in German. In France you could get away with Hello because the French "Allo" is very similar, but honestly you could just use Allo instead of Hello anyway and the readers will quickly guess the meaning based on context even with zero knowledge of French.
