To negate something, we use 不 (bù) before the verb, such as in this short dialogue:
A: 你是老师吗？= Are you a teacher?
B: 不是。= I'm not.
However, the verb 有 (yǒu) is the only one that requires 没 (méi) = 不是，不来，不忙, etc. vs 没有.
First of all, is this correct? If so, are there any social or historical (or other) reasons for this? What other uses does 没 have?
The Story of 没
As other commenters have noted, looking for logic in language is almost always futile. No natural language is logical. But there is a historical logic to language development; even if the existence of a phrase is a historical accident, it's sometimes interesting to see when that "accident" took place, and why.
Such is the case with 没. One interesting fact I learned from all this that contradicts what some others have said in their answers: 没 is not a contraction or shortening of 没有, at least not originally. MarkDBlackwell is right to point this out in his answer. The development of both words is slightly more complicated.
(All of the following comes from Xu Shiyi's excellent paper "否定词‘没’‘没有’的来源和语法化过程". To save time, I will not differentiate between places where I am quoting, summarizing, and expanding the original text. Apologies to the author; all errors are my own.)
The History of Negatives in Chinese
Before beginning, it's important to distinguish between two different categories of negative words: negative adverbs (否定副词) and negative verbs (否定动词). For our purposes, the former category is any word that can be put before a verb to negate its sense; the word 不 in modern Chinese is a good example. Negative verbs can stand alone at the "center" of a predicate (谓语). The word 没 is sometimes used in this way in modern Chinese. For example, in the sentence,
屋子里没人.
wūzi li méi rén
There is no one in the room.
没 is acting as a verb, not an adverb modifying another main verb. (The use of the word adverb is not exactly right here, by the way, because the category 副词 doesn't exactly correspond to the English category of adverbs, though it's pretty close.)
I'll also be making a distinction between the use of 没 as a "full-fledged verb" (corresponding to its reading "mò" today,meaning "to sink, to be covered") and its use as a negative verb.
Previously (in the oracle bone inscription days), Chinese had many negative adverbs, such as 不 、弗 、勿 and 毋, and later (during the Zhou and Qin dyansty period) added several more, like 非 、匪 、微 、无 、蔑 and 未. Both 无 and 蔑 were themselves negative verbs also, and both had the same meaning as 没有 does today.
没 on the Scene
The first textual evidence for the existence of 没 (as a verb meaning to sink/submerge) is in the Tang dynasty, but various pieces of evidence which I will not repeat here imply that 没 was already being used as a negative verb (not a negative adverb) before the Tang period.
The use of 没 as a verb meaning to sink into water gradually extended until 没 meant to disappear or lose. From a phonetic perspective, though, there is no link between the use of 没 as a full-fledged verb and its use as a negative. (Recall that two different pronunciations of 没 exist today which enforce precisely this difference.) Indeed, many experts have assumed that 没 originated as a negative word because it sounded the same as 未 (wèi, which is still used as a negative adverb today to mean "has not yet").
The author of this work, though, prefers to trace the connection between 没 and 无, which, as I pointed out above, has long had the same meaning as modern "没有."
Phonetic History of 没 and 无
Our knowledge of the phonology of middle Chinese comes mostly from a Song work called <广韵> Guangyun, which listed characters according to rimes [sic]. To indicate the pronunciation of a word, the dictionaries of those days used something called the 反切 method, which split a character up into two other characters, one representing the syllable onset and one representing the rime (rest of the syllable). Wikipedia's article on Fanqie quotes an example from Gari Ledyard of how such a system might work in English: we could encode the pronunciation of the word sough by using the two words sun and now, to show that the word starts with an "s" and ends with the same sound as the word "now." (I am omitting many details.)
These books also come with a table of rimes so that we know basically how the various consonants and vowels were articulated in those days. There were two different syllable onsets (called 明母 and 微母) which sounded similar (the former corresponds to modern initial "m", the latter variously to "w", "v" or "m"): 没 was 明母 (cf. modern méi), and 无 was 微母 (cf. modern wú). Various historical clues allow the author to conclude that the popular pronunciation of those two words (as opposed to the possibly literary pronunciation in the Guangyun) was the same. At that time, 无 also had the meaning of "to lose" when used as a full-fledged verb. This similarity, combined with the at-least similarity in sound, led to an eventual mixing of the uses of 没 and 无. By the Song dynasty, these two words were being used interchangeably. By the Yuan/Ming period, the process was completed, and 没 had replaced 无 as the negative verb meaning "没有."
But what about 没有?
The uses described above all still involve 没 as a negative verb. In fact, the above change also influenced the development of 没 as a negative adverb. 无 could always be used as a negative adverb in addition to being a negative verb, and the compound 无有 had always existed because 无 was viewed as being the direct negative of 有. As 没 gradually replaced 无, so too did 没有 replace 无有. (The form 不有 is attested, but was gradually superseded by 没有 as Chinese negative adverbs consolidated over time.)
There's more to be said about the process by which 没 and 没有 became fully gramaticalized, but I'm out of time and will have to come back later. Will be happy to post more if you're interested.
I have some examples:
我没开车
我没关灯
我没喝水
Also, you can say 我没关灯 by any chance is a shortening of 我没有关灯,but there is few people speak like that.
In some chances it is better to add 有 after 没, but in the other chances isn't.
有 can be translated as have in English. So when you want to say "I don't have money" in Chinese, you would say 我没有钱. If you don't have the word have in your English sentence, you don't need to use 有 in it's Chinese translation.
A couple of really common phrases that use the character 没:
没关系 = That's OK
没门 Literally means no door, but it actually means no way
没 has other means when it's pronounced as mò
沉没 = sinking
隐没 = disappear/hide
没落 = decline/downfall
没收 = confiscate
Firstly - you are correct, as are the others who have posted here. 没 is unusual in that it negates 有 and only 有，and for simplicity can itself serve as a contraction of 没有.
As others have pointed out, 没/没有 also has the unique grammatical role of indicating an action "not yet done" (in conjunction with 还 and 过) or "never done".
你去过香港吗？= Have you been to Hong Kong?
我还没去过。= I still have not gone.
我没去过。= I have never been.
As for why 有 takes 没, I think it's likely best to chalk that up to the irregularities of languages, which develop over time without any planning.
This is only a partial answer because I don't know all the details. I've been taught that when you use 没 with other verbs it indicates a different tense or time aspect.
我没去 (I have not gone)
我不去 （I do not go)
The reason the verb 有 (yǒu) takes 没 (méi) while other verbs take 不 (bù) is that 没 (méi) means, 'not have' and 有 (yǒu) means 'have'; thus they are tied linguistically (interestingly).
BTW, 没 (méi) can never literally serve as a contraction for 没有 (méiyǒu) 'not have', because by itself, historically, 没 (méi) already meant, 'not have'. One shouldn't discount the answer here by thinking of it as a contraction. Historically, this explains its peculiar attachment to 有 (yǒu).
If one realizes the historical nature of this question, it is good to refer to a dictionary of ancient Chinese, such as Analytic Dictionary of Chinese and Sino-Japanese by Bernhard Karlgren.
Perhaps your question really is how to have a feeling for when to use these two words in sentences?
没 (méi) means, 'not have'. It can be used alone, without 有 (yǒu), for that meaning. 没 (mò) has meanings related to that. It is a picture of a hand, a knife and water: a hand like a knife plunging into water: we don't see the hand anymore: 'not have'.
不 (bù) means, 'do not' (like, 'I do not', instead of a command). It is an (upside down) picture of a bird, landing. Perhaps it is the experience of seeing a bird not flapping wings, inactive?
Two concepts: 'do not' and 'not have'. They are not precisely the same concept, are they? If we think of 没 (méi) and 不 (bù) only as two negators, we are dropping what will help us learn them.
没 (méi) 'not have' and 有 (yǒu) 'have' in 没有 (méiyǒu) combine to mean 'not have', in a somewhat poetic way, like many other two-word combinations, for other linguistic reasons than meaning because 没 (méi) already means, 'not have'. If you think of 没 (méi) audibly as 'not have' and of 没有 (méiyǒu) as 'not have-have', you'll be all right.
Also, many verbs make sense both with 'not have' and with 'do not'. 我没开车 (wǒ méi kai-1 che-1) which glosses as 'I not have operate vehicle' (perhaps meaning, 'I haven't operated the vehicle'). 我不开车 (wǒ bù kai-1 che-1) glosses as, 'I do not operate vehicle' (perhaps meaning, 'I don't drive (vehicles)').
It's quite possible that you are asking a question that makes no sense in the context of the language itself. Asian languages tend to be less logically-oriented than Western languages, and more influenced by temporal events, like politics and history. Scholars, for example, are able to date documents by the conspicuous absence of certain characters - writing the emperor's name was forbidden in ancient China, so if we find that a strange character is being used instead of a more common-sense one that also happens to be in the name of a known emperor, we can say with reasonable conviction that this document was produced during that emperor's lifetime/reign. So, some questions, ESPECIALLY questions that start with the word "why," are more or less gibberish in the context of a lot of Asian cultures. I think you may have stumbled upon one such question here.
