bäuerlich
rural
Relating to farmers, agriculture, or life in the countryside; in the style of farmers.
Wir verbrachten den Urlaub in einer Pension mit bäuerlichem Charme.
We spent our vacation in a guesthouse with rural charm.

rural / peasant-like
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rural
Relating to farmers, agriculture, or life in the countryside; in the style of farmers.
Wir verbrachten den Urlaub in einer Pension mit bäuerlichem Charme.
We spent our vacation in a guesthouse with rural charm.

The adjective 'bäuerlich' (rural, peasant-like) is derived directly from the noun 'der Bauer' (the farmer). It describes things that are typical of farmers or country life.
Das Restaurant ist im bäuerlichen Stil eingerichtet.
The restaurant is decorated in a rural style.
In its neutral or positive meaning, 'bäuerlich' (rural) is often connected with aspects of country life. Common combinations are 'bäuerliche Idylle' (rural idyll), 'bäuerliche Produkte' (farm products), or 'ein bäuerlicher Betrieb' (a farm operation).
Auf dem Markt kann man frische bäuerliche Erzeugnisse kaufen.
At the market you can buy fresh farm products.
When 'bäuerlich' (peasant-like) describes behavior or manners, it has a derogatory meaning and means 'coarse' or 'uncultivated'. The context makes the negative judgment clear.
Sein lautes Rülpsen bei Tisch wurde als bäuerliches Benehmen empfunden.
His loud burping at the table was seen as peasant-like behavior.
fine
'fein' and 'bäuerlich' can form an opposite when it comes to appearance or behavior. 'fein' then describes something refined, well-mannered, or socially elegant. 'bäuerlich' can be meant derogatorily and then mean coarse, unpolished, or not refined. Important: 'bäuerlich' is not always negative. It can also simply have to do with farmers or agriculture. But as the opposite of 'fein', the derogatory meaning is intended.

Seine laute Art wirkte auf einige Gäste etwas bäuerlich.
His loud manner seemed somewhat peasant-like to some of the guests.
Bei der Veranstaltung trägt der Kunde einen feinen Anzug.
At the event, the customer is wearing a fine suit.