Particles НЕ and НИ

Probably, you have had some difficulties with the correct usage of particles НЕ [ne] and НИ [ni]. Firstly, it might seem that they sound in a similar way and secondly, both of them express negation. However, it's quite important to distinguish their meanings for the written language because the incorrect usage of a particle often changes the meaning of a sentence drastically. Well, here is the difference between them.

The particle НЕ [ne] expresses negation.
The particle НИ [ni] expresses the increase of negation.
Я не умею говорить по-русски - I can't speak Russian.
Дома ни души = Дома нет абсолютно никого - There is nobody at home.

We use the particle НЕ [ne] in the double negation construction (i.e. it expresses affirmation).
We use the particle НИ [ni] with a pronoun, when there is a negative predicate.
Не могу с вами не согласиться = Я согласен с вами - I agree with you.
Ни с кем не разговариваю - I don't speak with anybody.

The particle НЕ [ne] begins interrogative sentences.
The particle НИ [ni] begins the phrases expressing prohibition or absence of something.
Не правда ли, экзамен был очень сложным? - The exam was rather difficult, wasn't it?
Ни слова правды - Not a jot of truth.

We use the particle НЕ [ne] in exclamatory sentences.
We use the particle НИ [ni] in exclamatory senteces too but with the meaning of affirmation.
Чего мы только не слышали! - We've heard so much!
Чего мы только ни слышали, чего мы только ни видели = всё слышали, всё видели - We've heard so much, we've seen so much = we have heard everything, we have seen everything.

You should use these particles carefully not only in sentences but also in word-combinations (especially in set expressions) because the meaning often changes depending on chosen particle.
Не один = есть кто-то еще; несколько, много - not only = there is somebody else; several, many.
Ни один = никто - No one = nobody.