のIndicates possession

The particle の connects two nouns, usually to indicate possession or describe a noun. For example, 私の本 means 'my book.' When used with the pronoun の (meaning 'one'), the second の is dropped. For example, to say 'It's mine,' you would say 私のです, not 私ののです.

がある + Noun expresses possession, using a phrase with a noun, the particle が, and the verb ある to describe another noun (e.g., ピアノがあるレストラン = 'a restaurant with a piano'). In these cases, が can be replaced with の without changing the meaning (e.g., ピアノのあるレストラン).

の shows possession, like the English 's' (e.g., ケンジの本 = 'Kenji's book'). が marks the subject of a clause, often the doer of an action or the one with a certain trait. In modifying phrases, の and が are sometimes interchangeable without changing the meaning—for example, 先生が書いた本 and 先生の書いた本 both mean 'the book the teacher wrote.'

な-Adjectives describe nouns and connect to them using な (e.g., 好きな本 = 'a book I like'). の also connects nouns, with the first noun describing the second (e.g., 日本人の学生 = 'a Japanese student'). The choice between な and の depends on the type of word, though some words can function as both な-Adjectives or nouns with different nuances (e.g., 幸せな日 vs. 幸せの理由).