klassisch
classic
Typical, characteristic, or a textbook example of something.
Den Bus zu verpassen, weil man verschlafen hat, ist ein klassischer Montagmorgen.
Missing the bus because you overslept is a classic Monday morning.

classic / classical
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classic
Typical, characteristic, or a textbook example of something.
Den Bus zu verpassen, weil man verschlafen hat, ist ein klassischer Montagmorgen.
Missing the bus because you overslept is a classic Monday morning.

The adjective 'klassisch' (classic/classical) is closely related to the nouns 'die Klassik' (the classical period) and 'der Klassiker' (a classic example or a masterpiece).
Dieses Buch ist ein echter Klassiker der modernen Literatur.
This book is a true classic of modern literature.
You can recognize the exact meaning of 'klassisch' (classic/classical) from the noun it describes. 'Klassische Musik' (classical music) refers to a period, a 'klassischer Fehler' (classic mistake) is a typical mistake, and 'klassisches Griechenland' (classical Greece) means antiquity.
Ein klassischer Fehler ist, den Schlüssel im Auto zu vergessen.
A classic mistake is forgetting the key in the car.
In the meaning 'typical', 'klassisch' is often used in a slightly ironic or colloquial way to comment on a predictable, often negative situation.
Ich habe den Zug verpasst. Ein klassischer Start in die Woche!
I missed the train. A classic start to the week!
When talking about art, 'klassisch' (classical) is often combined with the art form itself. Typical collocations are 'klassische Musik' (classical music), 'klassische Literatur' (classical literature), or 'klassisches Ballett' (classical ballet).
Sonntagmorgens hört mein Vater gerne klassische Musik.
On Sunday mornings, my father likes to listen to classical music.
In the context of design, fashion, or architecture, 'klassisch' (classic) describes a timeless, elegant, and understated style that does not go out of fashion.
Für das Büro trägt er am liebsten klassische, dunkle Anzüge.
For the office, he prefers to wear classic, dark suits.
modern
'modern' and 'klassisch' can both describe a style, for example in clothing, furniture, music, or art. Often, however, they are opposites: 'modern' fits something new, current, or progressive. 'klassisch' is more suited to something traditional, timeless, or elegant. The words can be confused because both can sound positive. The difference is the perspective: 'modern' emphasizes the new, while 'klassisch' emphasizes what is tried and tested or timeless.

Ich trage heute einen klassischen schwarzen Anzug.
Today I’m wearing a classic black suit.
Die neue Küche ist hell, offen und sehr modern.
The new kitchen is bright, open, and very modern.