der Walzer
waltz
A social and competitive dance in triple time, characterized by the rotating movements of the couple.
Auf dem Ball haben sie die ganze Nacht Walzer getanzt.
At the ball, they danced waltzes all night.

waltz
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waltz
A social and competitive dance in triple time, characterized by the rotating movements of the couple.
Auf dem Ball haben sie die ganze Nacht Walzer getanzt.
At the ball, they danced waltzes all night.

The word 'Walzer' (waltz) comes from the verb 'walzen', which means 'to turn' or 'to roll'. This describes the characteristic turning movement of the dance.
Das Paar drehte sich elegant im Takt des Walzers.
The couple turned elegantly to the rhythm of the waltz.
The verb in the sentence reveals the meaning. If you 'tanzt' einen Walzer (dance a waltz), the dance is meant. If you 'spielt' (play), 'hört' (hear/listen to), or 'komponiert' (compose) it, the piece of music is meant.
Kannst du Walzer tanzen?
Can you dance the waltz?
The most common and important combination for the dance meaning is the fixed phrase 'Walzer tanzen' (to dance the waltz).
Auf ihrer Hochzeit tanzte das Brautpaar als Eröffnung einen langsamen Walzer.
At their wedding, the bride and groom danced a slow waltz as the opening dance.
The 'Wiener Walzer' (Viennese waltz) is an internationally known form of the dance and an important part of Austrian culture, especially during the ball season in Vienna.
Der Wiener Opernball wird traditionell mit einem Walzer eröffnet.
The Vienna Opera Ball is traditionally opened with a waltz.
polka
'der Walzer' and 'die Polka' are both dances and can also refer to pieces of music for these dances. For that reason, they can easily be confused, especially if you only know that both belong to dance music. The most important difference is in the rhythm and character: a waltz is in triple time and often feels rotating and flowing. A polka is in 2/4 time and is usually fast and lively. If it is about an elegant couple dance with turns, 'Walzer' fits. If the music seems faster and more folk-like, 'Polka' often fits.

Auf der Hochzeit tanzten Anna und Paul einen langsamen Walzer.
At the wedding, Anna and Paul danced a slow waltz.
Die Polka brachte sofort Bewegung in den ganzen Saal.
The polka immediately got the whole hall moving.