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「のです」・「のだ」 is the full version of the "explanatory extension."

「ん です」・「んだ」 is an extremely common spoken (and informally written) contracted variant. But, it can't be used directly after a noun, because then there wouldn't technically be a copula at the end of the sentence. So...

「なのです」・「なんです」(and 「なのだ」・「なんだ」) are used after nouns. Note that the な is actually a morphed だ, and you can see why it is used.

When asking for an explanation, obviously all of these things can have a か at the end. So something like:

アルバイトに行くのですか？　or （アルバイトに行くんですか？）
彼の自転車なのですか？　　　or （彼の自転車なんですか？）

When using the casual form in a question, though, the だ drops out. Notice that in this case you cannot contract の into ん:

アルバイトに行くのか？
彼の自転車なのか？

The above can sound somewhat abrupt (or masculine), so oftentimes people (especially, but certainly not limited to, women) will drop the か. The の is then said with the same rising intonation that the question marker か is usually given. Again, this cannot be contracted to ん:

アルバイトに行くの？
彼の自転車なの？

This form can also be used in non-question statements, in which case there would be no rising intonation on the の:

アルバイトに行くの。
彼の自転車なの。

Direct modification would be like 「アメリカの大学」 (American university), or 「四つ葉のクローバー」 (four-leaf clover).

Behaving more like [of] would be like 「世の中の男」 (the men of the world), or 「時間の問題」 (a matter of time).

Because of this [of] meaning, depending on the situation 「の」 might actually more closely correspond to other English prepositions, like [at], [in], or [on]:

「二階のエレベーター」 -- the elevator on the second floor (the second floor's elevator)
「公園のテーブル」 -- the table at the park (the park's table)
「本のページ」 -- the pages in the book (the book's pages)

Interestingly, while in English we have two separate ways to write the above examples, in Japanese they are both expressed simply with the particle 「の」.