der Bauer
farmer
A person who works in agriculture, cultivates land and/or raises animals; a farmer.
Der Bauer fährt am frühen Morgen mit dem Traktor auf das Feld.
The farmer drives the tractor onto the field early in the morning.

farmer / pawn
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farmer
A person who works in agriculture, cultivates land and/or raises animals; a farmer.
Der Bauer fährt am frühen Morgen mit dem Traktor auf das Feld.
The farmer drives the tractor onto the field early in the morning.

'Bauer' is a noun of the N-declension. This means that in all cases except the nominative singular, it takes the ending '-n' or '-en'. This is a common source of errors.
Der Bauer arbeitet auf dem Feld.
The farmer works in the field.
The meaning 'Spielfigur' (game piece) is only relevant in the context of games, especially chess. Words like 'Schach' (chess), 'Figur' (piece), 'ziehen' (to move), or 'opfern' (to sacrifice) point to this meaning.
Beim Schach ist der Bauer die schwächste Figur.
In chess, the pawn is the weakest piece.
In this use, 'Bauer' is a derogatory term and is used as an insult. It describes a person who is perceived as uncultured or impolite.
Er nannte ihn einen Bauern, weil er mit den Händen aß.
He called him an oaf because he ate with his hands.
farmer
'der Bauer' and 'der Landwirt' can both mean a person who works in agriculture, for example by cultivating fields or keeping animals. In this sense, they are very similar. 'Bauer' is the everyday word and is often used in conversation. 'Landwirt' sounds more official and modern, for example in news reports, forms, or professional texts. Important: 'Bauer' also has other meanings, for example in chess or as a derogatory term for a person. But in the relation to 'Landwirt', it is only about the person in agriculture.

Der Bauer bringt morgens frische Milch zum kleinen Laden im Dorf.
The farmer brings fresh milk to the small shop in the village in the morning.
Der Landwirt erklärt der Klasse, wie Kartoffeln angebaut werden.
The farmer explains to the class how potatoes are grown.