の Particle

In Japanese, the no の particle has several functions.

Creates Adjectives

The no の particle can create various sorts of adjectives, dubbed no の adjectives.

They're possessive adjectives, partitives, and appositives, created by marking nouns with the genitive case, or by acting as an attributive copula like the na な of na な adjectives.

neko no mimi
猫の耳
A cat's ears. (possessive.)
jinrui no go-wari
人類の５割
Five-tenths of the human race. (partitive.)
otouto no Sasuke
弟のサスケ
The younger brother, Sasuke. (appositive.)
futsuu no hito
普通の人
A person [that] is normal. (attributive copula.)

When it's a copula, it can be replaced by negative, past, past negative variants:

futsuu datta hito
普通だった人
A person [that] was normal.
futsuu denai hito
普通でない人
A person [that] is not normal.

Turns Words into Nouns

The no の particle is a nominalizer, which means it can turn words into nouns. This can happen for various reasons, and from this function of no の others functions are derived, so it's important to understand how it works.

First off, no の can generically replace any other noun, in which case it translates to "one" in English. Due to how plurals work in Japanese, it can also translate to "ones."

yasui hon wo katta
安い本を買った
[I] bought the cheap book.
[I] bought cheap books.
yasui no wo katta
安いのを買った
[I] bought the cheap one.
[I] bought the cheap ones.

To qualify no の with a na-adjective, you need the na な particle, which is an attributive copula, counterpart to the predicative copula da だ.

kono hon ga kirei da
この本が綺麗だ
This book is pretty. (predicative.)
kirei na hon
綺麗な本
A pretty book. (attributive.)
A book [that] is pretty. (as a relative clause.)
kirei na no wo katta
綺麗なのを買った
[I] bought the one [that] is pretty.

When it's qualified by a relative clause with a random verb, it can refer to the action expressed by that clause. In this case, no の can't really be translated to English. At best you can say it means "act," but that stops working with more complex clauses.

hashiru hito
走る人
A person [that] runs.
hashiru no ga tanoshii
走るのが楽しい
The act [that is] "to run" is fun.
Running is fun.
kimi ga naku no wo mita
君が泣くのを見た
[I] saw the act [that is] "you crying."
[I] saw you cry.
mono wo kau no ni okane ga kakaru
物を買うのにお金がかかる
For the act [that is] "to buy things," money costs.
It costs money to buy things.

When no の is combined with the ni に particle, as the compound particle noni のに, it sometimes expresses a secondary function, translating to "even though."

benkyou shiteiru no ni seiseki ga warui
勉強しているのに成績が悪い
Even though [I'm] studying, [my] grades are bad.

In this case, if you qualify the no の with an adjective instead of a verb, the no の no longer translates to "one." It simply refers to an adjective. Just like when it was a verb it referred to the act.

yasui no ni kaenai
安いのに買えない
Even though [it] is cheap [I] can't buy it.
- Here, no の isn't translated. Because its function is purely grammatical.

Note that, although the translation changes, the Japanese grammar syntax remains consistent. For example, with a na-adjective you still need the na な copula in order to qualify the no の.

kirei na no ni kareshi ga inai
綺麗なのに彼氏がいない
Even though [she] is pretty, [she] has no boyfriend.

In some cases, you'll need to say "even though A is N, B." Where N is a noun. Normally, to say "is [noun]" in Japanese you'd use the copula da だ.

isha da
医者だ
Is a doctor. (doctor is a noun.)

But the phrase above is predicative. We need to attribute no の, so we need an attributive copula. In which case, normally, when we need to qualify a noun with another noun, we use the no の particle as an attributive copula to create a no-adjective.

shokugyou ga isha da
職業が医者だ
The profession is doctor.
isha no shokugyou
医者の職業
The profession [that] is "doctor."
The doctor profession.

So, normally, we'd just replace that with the nominalizer no の.

*isha no no
~~医者のの~~
~~The の [that] is "doctor."~~

However, it turns out we don't do that. Instead, to qualify the no の nominalizer, you need to use the na な attributive copula instead. Even though we have a noun, not a na-adjective, we use the na な attributive copula.

isha na no
医者なの
The の [that] is "doctor."

Behold:

isha na no ni binbou da
医者なのに貧乏だ
Even though [she] is a doctor, [she] is poor.

The same thing principles applies when no の is combined with the particle de で, which then forms the compound particle node ので. This one has the primary meaning of "with," besides other things, and secondary meaning of "because."

To have an idea:

kieru inku de kaita
消えるインクで書いた
Wrote with ink [that] disappears.
kieru no de kaita
消えるので書いた
Wrote with the one [that] disappears.
(or...)
Wrote because [it] disappears.

Normally, however, it's the "because" meaning.

furui no de tsukaenai
古いので使えない
With the "old," not able to use.
Can't use with an old one.
Can't use when, if, it's an old one
Can't use because it's old.

Note that if the no の particle is after another particle, then it's turning the particle into an no adjective, not into a noun. But then again, no の only turns nouns into adjectives, and particles aren't nouns, I think, so maybe it's doing both things: turning them into nouns, and then into adjectives.

shachou kara no messeeji
社長からのメッセージ
The message from the company president.
- Here, shachou kara," from the company president," qualifies meseeji.
mirai e no tegami
未来への手紙
A letter toward the future.
- Here, mirai e, "toward the future," qualifies tegami.

The nominalizer no の can also come before the copulas da だ, desu です, dewanai ではない, and janai じゃない. In this case, it changes the meaning of the phrase. It can be used to assert an opinion, to give advice, or to explain something.

benkyou suru no da
勉強するのだ
benkyou suru no desu
勉強するのです
[You'll] study. (that's what you gotta do!)
- Literally:
- [It] is the の [that is] "to study."
akirameru no dewanai zo!
諦めるのではないぞ！
Don't give up!

In this case, the no の particle can contract into the n ん particle.

benkyou surunda
勉強するんだ
(same meaning.)
akiramerunjanai zo!諦めるんじゃないぞ！
(same meaning.)

Furthermore, it's also the source of the compound nanoda なのだ and its contraction nanda なんだ.

kirei na no da!
綺麗なのだ！
kirei nanda!
綺麗なんだ！
[It] is pretty!

Marks the Subject

In relative clauses, the no の can mark the subject instead of ga が.

okane ga aru hito お金がある人
okane no aru hito お金のある人
A person [who] has money.
okane ga nai hito お金がない人
okane no nai hito お金のない人
A person [who has] no money.

Asks Questions

As a sentence-ending particle, no の can ask questions like ka か.

iku ka? 行くか？
iku no? 行くの？
[Will you] go?

The main difference between no の and ka か question markers is that no の is more emotive. It tends to be used when you're asking something out of interest or out of worry. Meanwhile, ka か tends to be used in formal questions, toward customers, rhetoric questions, and so on.

They can be combined into no ka のか, which is used when expressing curiosity or doubt about the reason of something.

naze ame ga furu no ka?
なぜ雨が降るのか？
Why rain rains?
- ame 雨
  Rain. (noun.)
- furu 降る
  To rain. (verb.)
naze ame ga furu no ka wo kangaeyou
なぜ雨が降るのかを考えよう
Let's think [about] why rain rains.

Furthermore, in the end of sentences, it can also be combined with na な, to express more wonder about the question.

nani wo kangaeiteiru no ka na?
何を考えているのかな？
What is [he] thinking, [I wonder]?

This na な is a different na な from the attributive copula na な. Both na な can appear at once:

baka na no?
馬鹿なの？
Is [he] stupid?
baka na no ka na?
馬鹿なのかな？
Is [he] stupid, [I wonder]?

An observation: since you put the attributive copula na な before the sentence-ending no の, this implies that this no の is just the nominalizer no の but at the end of the sentence.

Also, both no の and ka か can list alternative questions in parallel.

iku ka ikanai ka?
行くか、行かないか？
iku no, ikanai no?
行くの、行かないの？
Will [you] go, will [you] not go? (which is it?! Decide already!)

When it comes right after the te-iru form of a verb, it can get contract to te-n-no てんの.

nani wo shiteiru no?
何をしているの？
What are [you] doing?
nani shite-n-no?
何してんの？
Wutch[u] doin'?

Asserts Statements

Also as a sentence-ending particle, no の can be used to assert a decision or to explain something. In this case, the intonation is lower, and it's used more by women than by men.

watashi wa kaeru no!
私は帰るの！
I'm going home! (I'm leaving! GOOD BYE TO YOU.)
- Not to be confused with:
- watashi wa kaeru no!
  私は蛙の！
  I'm a frog!

This can be combined with other particles, like no yo のよ. In which case it can be used to express one's opinion strongly, in order to correct someone about a misunderstanding, for example.