die Berühmtheit
fame
The state of being widely known and recognized by the public; renown.
Die junge Sängerin erlangte über Nacht weltweite Berühmtheit.
The young singer gained worldwide fame overnight.

fame / celebrity
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fame
The state of being widely known and recognized by the public; renown.
Die junge Sängerin erlangte über Nacht weltweite Berühmtheit.
The young singer gained worldwide fame overnight.

The word is made up of the adjective 'berühmt' (famous) and the ending '-heit'. The ending '-heit' turns adjectives into feminine nouns that describe a state or quality.
Aus dem Adjektiv „schön“ wird mit „-heit“ das Nomen „die Schönheit“.
With '-heit', the adjective 'schön' (beautiful) becomes the noun 'die Schönheit' (beauty).
You can recognize the meaning from the context: If the word is used uncountably for a state, it means 'Ruhm' (fame). If it is used countably for a person, often with the article 'eine' (a) or in the plural, it refers to a celebrity.
Sie strebte nach Berühmtheit. (Ruhm)
She strove for fame. (renown)
When 'Berühmtheit' (fame) refers to the state of being known, it is often combined with verbs such as 'erlangen' (to gain), 'genießen' (to enjoy), or 'zu Berühmtheit gelangen' (to achieve fame).
Durch den Film gelangte er über Nacht zu großer Berühmtheit.
Through the film, he achieved great fame overnight.
Only in the meaning 'bekannte Person' (well-known person) can the word appear in the plural: 'die Berühmtheiten' (the celebrities). The state 'Ruhm' (fame) has no plural form.
Zur Gala kamen viele Berühmtheiten aus Film und Musik.
Many celebrities from film and music came to the gala.
name
'der Name' and 'die Berühmtheit' can both have to do with famous people. 'Ein großer Name' (a big name) means a well-known and important person in a particular field, for example in music, sports, or science. 'Eine Berühmtheit' is a very well-known person; however, 'Berühmtheit' can also mean the state of being famous. Learners may confuse the words because both express recognition or fame. The difference is: 'Name' usually needs an addition here, such as 'großer' or 'bekannter', while 'Berühmtheit' by itself already refers to a famous person or to fame.

Nach dem Film wollten viele Fans ein Foto mit der Berühmtheit machen.
After the film, many fans wanted to take a photo with the celebrity.
Bei dem Konzert tritt heute ein großer Name der Jazzmusik auf.
A big name in jazz music is performing at the concert today.