1Used before a noun to ask what kind or what sort of (something) it is; 'what (kind of) ...'.
何の本を探しているの？
What kind of book are you looking for?
これは何の音だろう。
I wonder what kind of sound that is.
何の目的でここに来たのか教えてください。
Please tell me what purpose you came here for.
2Used in negative sentences as '何の＋noun＋も（ない）' to mean 'no...; not any; none at all'.
その計画には何の問題もない。
There are no problems with that plan.
彼には何の後ろめたさも感じられなかった。
He didn't seem to feel any guilt at all.
3Used to add emphasis to the preceding word or phrase, often appearing in the pattern '〜のなんの' to express astonishment, admiration, or emphasis.
あの試合の盛り上がりのなんの、会場は大変な熱気だった。
The excitement in that match was something else; the venue was full of energy.
彼の努力のなんのって、本当に尊敬に値する。
His efforts—I'm telling you—are truly admirable.
4(Interjection) A dismissive expression meaning 'It's nothing' or 'Don't worry about it,' used to downplay concerns or apologize.
A: 遅くなってごめん。 B: 何の、気にしないで。
A: Sorry I'm late. B: It's nothing, don't worry about it.
そんなことで落ち込むなんて何の、そんなに気にすることないよ。
Don't get upset over that—it's nothing; you don't need to worry so much.