の details
の — hangs one noun off another, the first describing the second
それ、私の傘じゃない？
sore, watashi no kasa ja nai?
Isn't that my umbrella? casual
昔からドイツの車が好きなんだ。
mukashi kara doitsu no kuruma ga suki na n da.
I've liked German cars for ages. casual
Examples
possession and belonging — the first noun owns the second or comes with it
それ、私の傘じゃない？
sore, watashi no kasa ja nai?
Isn't that my umbrella? casual
明日は会社の車で行きます。
ashita wa kaisha no kuruma de ikimasu.
Tomorrow I'll go in the company car. neutral
what kind — origin, subject or time, narrowing the noun that follows
昔からドイツの車が好きなんだ。
mukashi kara doitsu no kuruma ga suki na n da.
I've liked German cars for ages. casual
妹は日本語の先生です。
imouto wa nihongo no sensei desu.
My younger sister teaches Japanese. neutral
こちらが先週の会議の議事録でございます。
kochira ga senshuu no kaigi no gijiroku de gozaimasu.
These are the minutes of last week's meeting. formal
apposition — both nouns name the same person or thing, the first saying which one
友だちの田中さんが手伝ってくれました。
tomodachi no tanaka-san ga tetsudatte kuremashita.
My friend Tanaka helped me out. neutral
relative position — の hands the noun to 上[うえ], 中[なか], 前[まえ], 近[ちか]く and the rest of the position words
鍵、机の上に置いといたよ。
kagi, tsukue no ue ni oitoita yo.
I've left the key on the desk. casual
駅の近くにコンビニがあります。
eki no chikaku ni konbini ga arimasu.
There's a convenience store near the station. neutral
Formation
{N}のattaches to noun格助詞＋の（母への手紙・東京からの電車・友だちとの約束）attaches to noun phrase (schematic)
Notes
×静かの部屋. The head noun drops when it is recoverable, leaving の
standing for it: これは私のです.
Related patterns
Similar
Both put a topic in front of a noun, but の only links the two and leaves the relation to be guessed (日本の本 is a book from Japan or about it), while についての pins it to "about" — which is why a title that must not be misread spends the extra syllables on についての.
Lookalikes
Same kana, different job — between two nouns の links them (私の本), at the end of a finished sentence it is the explanatory particle asking or accounting for something (どうしたの？), and no noun follows it there at all.
The の of 〜のが好きだ is the nominaliser, which manufactures a noun out of a whole clause (泳ぐのが好きだ); the genitive の links two words that were nouns to begin with, so what precedes it is the test.
What precedes decides — the genitive links two words that were nouns already (会社の車), the nominalizer takes a whole clause and manufactures a noun out of it (毎朝走るの).