A lot of our customers that came to China have asked me "how do you answer no to a question?". Well I couldn’t give a straight answer, because there’s no yes or no like in English. It always needs to be followed by something else...

The explanation in the dictionary would be 不(bú) or 否 (fǒu). But it really depends on how you use it.

For example:
Q: Did you speak to him today? 你今天跟他说话了吗？nǐ jīntiān gēn tā shuōhuà le ma
A: No 没有 méiyǒu
Meiyou means (to) not have

Q: Do you have apples? 你有苹果吗？nǐ yǒu píngguǒ ma?
A: No 没有 méiyǒu
Same like the above example, no will be "no have". This no answer is given when somebody asks you if you did something or have something.

Q: Do you want to eat apples? 你要吃苹果吗？nǐ yāo chī píngguǒ ma?
A: No 不要 búyào
This means Not Want. It’s used when somebody is asking you if you want something or basically when asking about something that is happening in present

Q: Is this your friend? 这个是你朋友吗？ zhègè shì nǐ péngyou ma?
A: No 不是 bú shì

Q: Do you know his name? 你知道他的名字吗？ nǐ zhīdào tāde míngzi ma?
A: No 不知道 bù zhīdào
This means I don’t know. 不 bù: no 知道 zhīdào: (to) know

The other example, 否 (fǒu) is not used so much or at all in spoken language. It is used more in writing.
Q:Do you confirm? 您是否确认？ nín shìfǒu quèrèn?
A: No 否 fǒu

And there you have it, all ways to say no in Chinese. Which way you use to say no really depends on the contents.