die Manier
manner
The characteristic way in which a person does something or behaves.
Seine arrogante Manier, mit anderen zu sprechen, machte ihn unbeliebt.
His arrogant manner of speaking to others made him unpopular.

manner / manners
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manner
The characteristic way in which a person does something or behaves.
Seine arrogante Manier, mit anderen zu sprechen, machte ihn unbeliebt.
His arrogant manner of speaking to others made him unpopular.

Pay attention to the difference between singular and plural, because the meaning changes significantly. 'Die Manier' (singular) describes a specific way or manner, while 'die Manieren' (plural) refers to good or bad manners.
Seine Manier, den Kopf schief zu halten, war sehr charakteristisch.
His manner of holding his head at an angle was very characteristic.
In the singular, 'Manier' (manner) often describes a way of doing something that is perceived as particular, peculiar, or even slightly negative. It is not as neutral as 'Art' (way/type) or 'Weise' (way/manner).
Ich mag seine belehrende Manier überhaupt nicht.
I really don't like his lecturing manner at all.
When referring to good behavior, the word is almost always used in the plural. Typical phrases are 'gute/schlechte Manieren haben' (to have good/bad manners) or 'jemandem Manieren beibringen' (to teach someone manners).
Er hat wirklich schlechte Manieren, er unterbricht die Leute ständig.
He really has bad manners; he constantly interrupts people.
In the context of art or literature, 'Manier' (style/manner) is used with the preposition 'in' to describe a style. The structure is often 'in der Manier von' (in the manner of), followed by a name or an era.
Das Gebäude wurde in der Manier des Barock errichtet.
The building was constructed in the manner of the Baroque.
kind / way
'die Art' and 'die Manier' can both describe how someone does something. 'die Art' is neutral and very general. You can use it for actions, behavior, or methods. 'die Manier' refers more strongly to a person's personal, typical way and often sounds somewhat more formal or even slightly critical. If you simply want to say how something is done, 'Art' is usually the better fit. If you want to emphasize a person's particular style or behavior, 'Manier' is more suitable.

Herr Lange erklärt alles in seiner eigenen Manier und macht dabei viele Pausen.
Mr. Lange explains everything in his own manner and takes many pauses while doing so.
Die ruhige Art der Lehrerin hilft den Schülern bei schwierigen Aufgaben.
The teacher's calm manner helps the students with difficult tasks.