の
no
の is not 'of'. It is a hook.
The picture A small hook that hangs the first word onto the second. It only tells you what belongs to what.
Where the picture shows up in の
-
whose it is
これは私の傘です。
kore wa watashi no kasa desu
This is my umbrella.
-
what kind it is
日本語の先生です。
nihongo no sensei desu
He's a Japanese teacher.
-
the person is the post
部長の田中です。
buchō no Tanaka desu
I'm Tanaka, the department head.
部長の田中 is not the department head's Tanaka — it is Tanaka who is the department head. You meet this on every business card.
-
an action turned into a thing
歩くのが好きです。
aruku no ga suki desu
I like walking.
-
the 'one' whose name is left out
赤いのをください。
akai no o kudasai
I'll take the red one, please.