Bù (不) and Méi (没) are both negative words. I listed several situations to help you use them properly. But remember, there are always overlapsand exceptions.

1. For expressing "don't have" and "there isn't or there aren't", you must use "méi yǒu" or "méi ". You can not use "bù". "méi "is the shorter term for "méi yǒu", usually you can replac each other.

For example:
wǒ méiyǒu shíjiān. 我没有时间。
wǒ méiyǒu qián. 我没有钱。
wǒ méi kòng . 我没空。
yīngguó méiyǒu dàxióngmāo. 英国没有大熊猫(giantpanda)。

2. "bu" is usually used to describe the change of the state of something, but "mei" is usually to describe a negative fact.

a) chē bù zǒu le . 车不走了。(The car was going before, but now it is not going any longer.)  
chē méi zǒu. 车没走。（The car is there. It hasn't gone yet.）

b) tiān bú xiàyǔle. 天不下雨了。(It was raining before, but now it is not raining any longer.)
tiān méi xiàyǔ. 天没下雨。(It hasn't rained. It didn't rain.)

3. When you describe somthing that hasn't taken place, you can use "méi yǒu".

For example:
tā méiyǒu lái . 他没有来。
wǒ méiyǒu chīfàn . 我没有吃饭。
wǒ méiyǒu shōudào tāde diànzǐyóujiàn. 我没有收到他的电子邮件。

4. When you describe something has taken place and the change of state which is a negative one, such as disappeared, you use "méi".

For example:
píjiǔ méiyǒu le. 啤酒没有了。(In this context, we usually use "le" in the end of the sentence.)

5. When it is a personal decision and willingness, you use "bù".

For example:
tā bù lái le . 他不来了。(It means he has decided not to come.)
tā méi lái. 他没来。(It just describes the fact that he didn't come.)
wǒ bù chīfàn . 我不吃饭。(It means I don't want to eat.)
wǒ bù xǐhuān měiguó píjiǔ. 我不喜欢美国啤酒。 (It means I don't like to drink American beer.)

6. When you give a negative judgment to the state, attribute or manner of a certain thing or a fact, you use "bù".

For example:
wǒ juéde měiguó de píjiǔ bùhǎo . 我觉得美国的啤酒不好。
tā xiěde zì bùhǎo. 他写的字不好。