'Do' and its various forms (does, did, don't, doesn't, didn't) are some of the most confusing words in the English language. Those of us who grew up speaking English use them effortlessly, but for anyone learning English as a second language 'do' is ridiculously complex.
This is largely because the word 'do' does work both as a standard verb and as a somewhat random grammatical marker, and many of the jobs that it does as a grammatical marker are not found in any languages other than the Celtic languages, and moreover, are somewhat inconsistent.
The basic verb: 'do' is used as a verb indicating the performance of an action or activity. "I do a lot of stuff", "He did the deed". 'Do' is the catch-all word used when no more specific word is clearly more appropriate; for example, one would not usually say "I did the race", but rather "I ran the race".
The idiomatic verb: in addition, there are a whole lot of idiomatic phrases that use 'do' regardless of whether there is a more appropriate verb. For example, it is common to say "I'll do the dishes" rather than "I'll wash the dishes"; this is an idiomatic because without prior knowledge of common English usage there is no way to know if "doing the dishes" means setting the table, putting food on plates, putting away the dishes, or perhaps washing the dishes. Other idiomatic usages of this sort include "do your nails", "do lunch", and "that does it".
The negative form: In regular usage, negative sentences may be formed with the addition of 'don't' (short for 'do not'); "I don't want fish". ("I do not want fish" is also correct, but may be seen as more formal or more emphatic). There are other words that perform this function, and work in exactly the same way: can't (cannot), won't (will not), and shouldn't (should not).
However, when making a negative form of a sentence containing 'do', one must keep the original 'do'; one must say "I don't do a lot of stuff", and cannot say "I don't a lot of stuff". This is true regardless of the conjugation: "he doesn't do the deed"; "he didn't do the deed". This also holds when negating 'do' sentences with other negative forms: "I can't do that".
The light verb: 'do' is frequently used paired with a noun to form a verbal noun. For example, "he did work", "I did a study on fish". These constructions can almost always be made without the light verb; "he worked", "I studied fish". However, use of the light verb may indicate that the action is a single instance rather a series of actions or an ongoing action; compare "I did a dance" to "I danced".
In some cases this is a matter of style; for example, "I work on the work" is a perfectly clear statement in English, but in order to avoid repeating the word 'work' we use 'do' as a light verb; "I do the work". However, this is often a matter of personal preference; for example, "I did the painting" is not significantly more common than "I painted the painting".
The auxiliary verb: certain constructions require the use of an auxiliary verb. The most common of these is in asking yes/no questions. In English, these questions require a subject-auxiliary inversion -- meaning that the question must start with an auxiliary verb. There are comparatively few words we can use for this purpose, including 'are', 'have', and 'do'. ("Are you busy?"; "Have you won?"; "Do you want a fish?")
This is a fairly hard thing for ESL learners to pick up, particularly because very few languages do this -- in fact, in Europe it is only the Celtic languages and a few dialects of Italian (and it is much less common, perhaps even unheard of, outside of Europe). In most languages, one simply asks "you want a fish?", and in fact, this form is commonly used in informal English. But formal/proper English absolutely requires that a yes/no question starts with an auxiliary verb, even if the root sentence does not require one (e.g. both "I want a fish" and "I do want a fish" are proper English).
Things become even more complicated when looking at wh- questions. A wh- question does not require subject-auxiliary inversion if the the wh- word is the subject (or part of the subject); e.g., "what is this?", "where are you?". But if the wh- word is not the subject, the question is required to have an auxiliary verb immediately after the wh- word; e.g. "where will you go?", "what do you do?".
'Do' is also used in the anaphora of verb-phrase ellipsis, in which a second instance of a verb is replaced with 'do' in sentences comparing two things; e.g. "he works more than I do". This is generally the default grammar for this sort of sentence in English, as using the same verb twice in a sentence would sound odd. However, the rules for using 'do' to replace the second occurrence of a verb are complex; for example "he works and I do" is incorrect, but "he works and I do too" and "he works and I don't" are both correct. However, these rules are not specific to 'do', and will also apply to similar cases of ellipsis using 'can', 'will', 'should', and others.
And to tops things off, when 'do' is used as an auxiliary verb in a simple sentence, it most often serves to indicate emphasis; "I do want a fish" is more emphatic than "I want a fish".
It is worth noting that technically, the 'do' part of 'don't', used to indicate negative form, is also an auxiliary verb, and is often considered 'meaningless' in the same way as the other uses of auxiliary do. Most European languages will have negative constructions of the general form "I not want fish".
Despite this mess, grammar is one of those things that native speakers pick up as young children with comparatively little effort, and next to no awareness. While some of the rules outlined above appear incomprehensible to non-native speakers, they are so obvious to native speakers that outlining them as 'rules' seems to confuse matters rather than clarify them. This means that many native English speakers are unable to help foreign speakers correct their grammar -- they know what is right, but can't explain it.
There is some hope that as English becomes a global language it may simplify some of its more needlessly complex rules, but the grammar reform movement is dwarfed by even the minuscule spelling reform movement. But even without formal recognition some rules are changing, particularly the dropping of auxiliary verbs in asking yes/no questions in informal speech. Y'know?