能 (néng) is another Chinese word that is often translated as "can." However, it has a slightly different meaning than the others. 能 is used to express ability or possibility of doing something. This article will further explain its use.

Structure

Expressing Ability
The verb 能 is used to express either ability or possibility.

Examples
能 indicates ability when used with activities that are not consciously learned or studied.

Wǒ nénɡ chī sìshí ɡè jiǎozi.
我 能 吃 四十 个 饺子。
I can eat 40 dumplings.

Expressing Possibility
Examples
When used with activities that are consciously learned or studied, 能 generally means that circumstances do not allow execution of the action. In other words, it's not possible. Sometimes this is a bit of an exaggeration and actually just means it's "not allowed."

Shànɡkè de shíhou bú nénɡ shuō yīnɡwén.

上课 的 时候 不 能 说 英文。

 
(my teacher has forbidden me from speaking English.)In class, you can't speak English.

Other usages

From these examples we can see such circumstances might be pertaining to getting someone's consent or reliant on the speaker's health. 会 can be used instead if the speaker wishes to express that they are able to execute an action requiring a consciously studied skill.

Because 能 can express possibility, it is often used to form polite questions, something like "would it be possible" in English:

Nǐ nénɡ bú nénɡ ɡěi wǒ nà běn shū?
你 能 不 能 给 我 那 本书？
Can you give me that book?