jucken
to itch
To cause an unpleasant feeling on the skin that creates the urge to scratch.
Der Mückenstich auf meinem Arm juckt unerträglich.
The mosquito bite on my arm itches unbearably.

to itch
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to itch
To cause an unpleasant feeling on the skin that creates the urge to scratch.
Der Mückenstich auf meinem Arm juckt unerträglich.
The mosquito bite on my arm itches unbearably.

The grammar reveals the meaning. If a body part is the subject, it means 'to itch'. The fixed negative phrase 'Es juckt mich nicht' always means 'I don't care'.
Mein Rücken juckt schrecklich.
My back itches terribly.
To describe an itch, the body part can be the subject. Alternatively, you can use the impersonal form 'Es juckt mich an/auf...' (It itches me on...).
Mein ganzer Arm juckt.
My whole arm itches.
The colloquial meaning is used in a fixed structure: 'Es juckt + person in the accusative + (nicht)' (It does/doesn't matter to [person]). The subject is always 'es'.
Es juckt ihn nicht, dass er zu spät ist.
He doesn't care that he's late.
The meaning 'to interest' or 'to matter' is very informal and is mainly used in spoken language. It often expresses indifference.
Ganz ehrlich? Deine Meinung juckt mich nicht die Bohne.
Honestly? I couldn't care less about your opinion.
to interest
'interessieren' and 'jucken' can both express that something is important or interesting to someone. 'interessieren' is neutral and fits almost all situations: you can be interested in a topic, or a topic can interest someone. In this meaning, 'jucken' is colloquial and is almost always used in negative sentences, for example 'Das juckt mich nicht' (I don't care about that). It then means: I don't care. Learners may confuse the words because both have to do with interest. As a practical rule: use 'interessieren' for normal interest. Use 'nicht jucken' for casual, rather direct indifference.

Die Meinung der anderen juckt ihn nicht.
He doesn't care about other people's opinions.
Im Kurs interessieren wir uns besonders für die Termine der Prüfung.
In the course, we are especially interested in the exam dates.