der Pelz
fur
The dense, soft coat of hair of certain mammals, which protects them from the cold.
Der Bär hat einen dicken Pelz, der ihn im Winter wärmt.
The bear has thick fur that keeps it warm in winter.

fur
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fur
The dense, soft coat of hair of certain mammals, which protects them from the cold.
Der Bär hat einen dicken Pelz, der ihn im Winter wärmt.
The bear has thick fur that keeps it warm in winter.

To distinguish the meanings, pay attention to the context. If someone is talking about an animal, it refers to its coat of hair. If it is about clothing that a person wears, it means a garment.
Der Pelz des Fuchses ist im Winter besonders dicht.
The fox's fur is especially thick in winter.
In the fixed expression 'jemandem auf den Pelz rücken' (to crowd someone / to get too close to someone), the person being pressured is always in the dative. The structure is therefore: [person in the dative] + 'auf den Pelz rücken'.
Die Verkäufer im Basar sind mir zu sehr auf den Pelz gerückt.
The vendors at the bazaar crowded me too much.
When referring to the coat of hair of animals, 'Pelz' (fur) is often combined with adjectives such as 'dicht' (dense/thick), 'weich' (soft), or 'warm' (warm).
Der Biber hat einen sehr dichten und wasserabweisenden Pelz.
The beaver has very dense and water-repellent fur.
In connection with clothing, the combinations 'einen Pelz tragen' (to wear fur) and the distinction between 'echtem Pelz' (real fur) and 'künstlichem Pelz' (artificial fur) or 'Kunstpelz' (faux fur) are very common.
Sie trägt aus Überzeugung nur künstlichen Pelz.
Out of conviction, she only wears faux fur.
hair
'das Haar' and 'der Pelz' can both have to do with animal hair. 'Haar' is the general word for a single hair or for hair in general. 'Pelz' means the dense, soft coat of certain animals, especially when it protects them from the cold. 'Pelz' can also mean a piece of clothing made from animal fur. The words can be confused because fur consists of hairs. But for people, you normally say 'Haar', not 'Pelz'. For animals, 'Pelz' more often describes the whole warm, dense layer.

Der Fuchs hat im Winter einen dicken Pelz.
The fox has thick fur in winter.
Timo findet ein helles Haar auf seiner Jacke.
Timo finds a light-colored hair on his jacket.