der Klassiker
classic
A work of art, music, or literature that is considered timeless, exemplary, and of lasting value.
Beethovens 5. Sinfonie ist ein Klassiker der Musikgeschichte.
Beethoven's 5th Symphony is a classic of music history.

classic
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classic
A work of art, music, or literature that is considered timeless, exemplary, and of lasting value.
Beethovens 5. Sinfonie ist ein Klassiker der Musikgeschichte.
Beethoven's 5th Symphony is a classic of music history.

'Der Klassiker' (the classic) is the noun related to the adjective 'klassisch' (classic/classical). Something that is 'klassisch'—that is, timeless, typical, or exemplary—is 'ein Klassiker' (a classic).
Dieses Automodell hat ein klassisches Design.
This car model has a classic design.
The meaning of 'Klassiker' becomes clear from context. If it's about art, literature, or music, it refers to a masterpiece. In a sports context, it is an important match. In everyday life, it describes a typical situation.
„Faust“ von Goethe ist ein Klassiker der deutschen Literatur.
'Faust' by Goethe is a classic of German literature.
To specify what kind of classic is meant, the field is often named: 'ein Klassiker der Filmgeschichte' (a classic of film history), 'ein Literaturklassiker' (a literary classic), or 'ein Rock-Klassiker' (a rock classic).
Dieser Song ist ein echter Klassiker aus den 80er Jahren.
This song is a real classic from the 80s.
The meaning 'typical situation' is colloquial. It is often used with a certain irony to talk about a recurring mishap or a predictable action.
Ich habe schon wieder den Bus verpasst – der Klassiker am Montagmorgen.
I missed the bus again—the classic Monday morning situation.
institution
'die Institution' and 'der Klassiker' can both refer to something that is very well known and has long played an established role. 'die Institution' fits more with a person, thing, or organization that is a fixed part of a community. 'der Klassiker' more often means a work, a product, or also a typical situation that keeps recurring or is considered timeless. You can confuse the words when you want to say that something is well known and proven. A practical tip: A person or place can be an institution within a group; a film, a book, or a typical mistake is more likely a 'Klassiker'.

Dieser alte Film ist ein Klassiker, den viele Familien jedes Jahr sehen.
This old film is a classic that many families watch every year.
Der kleine Laden an der Ecke ist in unserer Nachbarschaft eine echte Institution.
The little shop on the corner is a real institution in our neighborhood.