
geläufig
common / familiar
Generally known, familiar, or widespread.
Sein Name ist mir geläufig, aber ich kann kein Gesicht zuordnen.
His name is familiar to me, but I can't put a face to it.
common / fluent
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common / familiar
Generally known, familiar, or widespread.
Sein Name ist mir geläufig, aber ich kann kein Gesicht zuordnen.
His name is familiar to me, but I can't put a face to it.

fluent
Fluent, effortless, and confident, especially with regard to language or an activity.
Nach einem Jahr im Ausland war ihm die Sprache sehr geläufig.
After a year abroad, he was very fluent in the language.
The adjective 'geläufig' (common/fluent) is related to the verb 'laufen' (to run). It describes something that 'runs' well, meaning is fluent, or that is figuratively in circulation and therefore known to many people.
Diese Redewendung ist heute nicht mehr so geläufig.
This idiom is no longer so common today.
When it is about knowledge (a name, a term), 'geläufig' (common/familiar) usually means 'bekannt' (known). When it is about a skill (a language, an instrument), it means 'fließend' (fluent).
Die Abkürzung „z.B.“ ist jedem Deutschlerner geläufig.
The abbreviation 'z.B.' is familiar to every German learner.
In the meaning 'bekannt' (known/familiar), 'geläufig' is almost always used with a person in the dative. The structure is: Something is [mir, dir, ihm, ihr...] geläufig.
Der Name der Autorin ist mir zwar geläufig, aber ich habe noch kein Buch von ihr gelesen.
The author's name is familiar to me, but I haven't read any of her books yet.
In the meaning 'fließend' (fluent), 'geläufig' can also be used as an adverb to describe how well someone performs a skill. It then stands directly with the verb.
Er kann das Musikstück schon sehr geläufig spielen.
He can already play the piece of music very fluently.
known
Bekannt and geläufig can refer to a name, term, or expression that people know. Bekannt generally says that it is familiar to many people; geläufig emphasizes more strongly that it is present in people’s memory or in common usage. Use “bekannt” for general familiarity and “geläufig” for familiar usage.

Der Name des neuen Bürgermeisters ist den Einwohnern inzwischen sehr geläufig.
The new mayor’s name is now very familiar to the residents.
Der Name des neuen Bürgermeisters ist inzwischen in der ganzen Stadt bekannt.
The new mayor’s name is now known throughout the city.