Bu and mei(you)

Bu and mei(you) can both be used to negate verbs, however they are used differently. Let's first look at how they are used for statements in the past and present.

Past and present statements

Perhaps the most intuitive explanation of the difference between 不 and 没（有）is that 不 negates actions in the present whereas 没 negates actions in the past. For example:

我不去 (I'm not going)
他没去 (I didn't go)
我没告诉他！ (Honestly, I didn't tell him)
我不告诉你 (I'm not telling you)

At first this is a good way to remember the difference between the two, but this explanation doesn't neatly cover all uses.

Personal and factual statements

A slightly less intuitive but more encompassing explanation is that 不 is used to negate statements which are based on personal circumstances, whereas 没 is used to negate statements of fact.

For example:

我没去 (I didn't go – an objective statement not reliant on personal views, it is a historical fact)
我不去 (I won't go – a decision based on personal circumstances. I won’t go because I don't have the money, because I'm busy or so on. This decision is up to me)
我今天没吃饭 (I haven't eaten today. This is a simple objective statement)
我今天不吃饭 (I'm not eating today. This is an opinion based on personal reasons; perhaps the speaker is feeling ill, is too busy or is too lazy. Again, this is my decision)

With this clear, it is now easier to explain another two uses of 不 and 没有

First, 没 is always used to negate 有. 有 means to be exist; when we state that things exist in everyday life, it is a factual statement (for example, "饼干没有了" "there are no biscuits" is a fact . For this reason we cannot negate 有 with 不.

Second, when we make negative judgments about things, we use 不, as this is a personal statement. Negating adjectives generally involves making a judgement, which is a personal statement, therefore we do not use 没 to negate adjectives. We can only say "不好" or "不漂亮", not "没好" or "没漂亮".

The examples below are a little more difficult, as these statement are very similar. However the same rule applies:

水没凉 (the water hasn't cooled down yet – this emphasizes that the objective process of water cooling hasn't finished yet, as it hasn't yet reached room temperature)
水不凉 (the water isn't cool – here the speaker is making statement based on their belief of what cool is. I might think this water is cool, but you may not, the standard we use is subjective)
我不是这个意思 (No, you've got me wrong, you've judged the situation wrongly - a personal judgement on my behalf that the listener has misjudged me)
我没有这个意思 (I never implied that - an objective fact)