die Art
way
The specific way or method in which something is, happens, or is done.
Ich mag ihre freundliche Art zu sprechen.
I like her friendly way of speaking.

kind / way
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way
The specific way or method in which something is, happens, or is done.
Ich mag ihre freundliche Art zu sprechen.
I like her friendly way of speaking.

'Art' is a common false friend. It does not mean the English word 'art' (in German: die Kunst), but rather 'Art/Weise' (kind/way), 'Sorte' (sort), or 'Typ' (type).
Diese Art von Blumen habe ich noch nie gesehen.
I have never seen this kind of flower before.
To describe a method or behavior, 'Art' (way) is often used in the constructions 'die Art, wie ...' (the way in which ...) or 'die Art zu + infinitive' (the way of + -ing).
Mir gefällt die Art, wie sie ihre Geschichten erzählt.
I like the way she tells her stories.
When asking about a sort or type, the question 'Welche Art von ...?' (What kind of ...?) is very common. The answer often also contains 'eine Art von ...' (a kind of ...).
Welche Art von Büchern liest du am liebsten?
What kind of books do you like reading most?
The expression 'auf eine Art' (in a way) or 'auf eine Art und Weise' (in a certain way) means 'somehow' or 'in a certain respect'. It is used to soften a statement somewhat or to express a nuanced perspective.
Auf eine Art kann ich verstehen, warum er so gehandelt hat.
In a way, I can understand why he acted like that.
You can easily recognize the biological meaning 'Spezies' (species) from context. If the topic is animals, plants, or terms like 'ausgestorben' (extinct), 'bedroht' (endangered), or 'heimisch' (native), this is the meaning intended.
Der Eisbär ist eine durch den Klimawandel gefährdete Art.
The polar bear is a species endangered by climate change.
family
'die Art' and 'die Familie' can both refer to groups, but they belong in different contexts. 'die Art' often means a kind or type; in biology, it is a precise group of living beings that can reproduce with one another. 'die Familie' in everyday language means parents, children, and relatives. In biology, it is a larger group than the species. Learners may confuse the words because both can talk about groups of people, animals, or plants. As a simple guide: for relatives, you say 'Familie'; for a kind, type, or biological species, you say 'Art'.

Diese Art von Vogel lebt nur in warmen Regionen.
This species of bird lives only in warm regions.
Vor der Reise besucht Mehmet noch seine Familie.
Before the trip, Mehmet visits his family.