Is used to denote something as new information.

んだ is normal. のだ is very assertive. Used in 論文 or formal writing etc.

Telling someone something they didn't know

オランダ人なんです。 I'm dutch (if you didn't know)

ところで、ニックはヴィガンなんです。 Nick is vegan btw (people didn't know)

車は買いません。高いんだ I won't buy a car. That is, because it is expensive.

実は、昨日人を殺したんだ… Actually... Yesterday I killed someone...

今日は寝ません。勉強するんです。 I won't sleep today. It is because I (will) study.

Explaining circumstances to clarity a situation

私は学生なので、お金がないです Because I'm a student, I don't have money

Scenario: You're seen with your sister. Oh no, they might get the wrong idea. 彼女じゃないんです。妹なんです。

(actually, )She's not my girlfriend, she's my my sister

Also often used in conjunction with Introductory remark to make an introductory remark

駅に行きたいんですが、生き方を教えてもらえませんか？ I want to go to the station, which way do I need to go?

仕事を辞めます。新しい就職するんだ I quit my job. In other words, I will find a new job.

Infeminine speech pattern it is common to drop だ. This is also the case here. It is alsocasual.

空っぽんだ (male) 空っぽなの (female)

私は日本人なんだ。(male) 実は日本人なの (female)

When used in a question, implies a that you're curious and want the other to explain.

何をしていますか？ ↔ 何をしているんですか？ What are you doing?　 ↔ What are you doing? (tell me, I'm curious!)

犬が好きなの。 Do you like dogs? (said after e.g seeing someone pet dogs)

明日は勉強するんですか。 Are you studying tomorrow? (seeing someone who didn't study today)

If you're in disbelief, or are surprised, んだ is your friend!

今{蜘蛛|くも}を食べたんですか？ did you just eat a fucking spider? ಠ_ಠ

えええもう帰るんですか？ You're already going home? (I really thought you were gonna stay longer)

In questions, んだ cannot be used. So, んです is the formal, and の is used for bothmale andfeminine

何を~~しているんだ~~？ ✕ 何をしているの？ ○ 何をしているんですか？ ○