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. の hooks a noun onto a noun; な hooks a quality onto one — 日本語の先生 against 親切な先生.
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.
A name tag dangling from the handle of the umbrella
-
what kind it is
日本語の先生です。
nihongo no sensei desu
He's a Japanese teacher.
A badge reading 'Japanese' hanging on the teacher's chest
-
the person is the post
部長の田中です。
buchō no Tanaka desu
I'm Tanaka, the department head.
A business card with 'department head' and 'Tanaka' stacked one above the other
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 act of walking packed into a box and set on the shelf marked 'things I like'
-
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.
Picking the red one off the shelf without ever saying what it is
All of it in one picture
の is a small hook that hangs one word onto the next. Hang your name tag on the umbrella and it is 'my umbrella'; hang a 'Japanese' badge on the teacher and it is 'a Japanese teacher'; hang 'department head' on Tanaka and the man is the post. Hook the act of walking and it becomes a thing you can like; drop the name and keep the hook and you get 'the red one'. So の does not make meaning — it only tells you what is hanging on what.