Modern English seems to require this verb in several circumstances, where most other European languages don't seem to need it. (See? I just used it.)
For example, in questions: "Do you have a dog?" Whereas, "Have you a dog?" would be normal in other languages or in the English of days gone by.
Another example: "I do not know him." Again, "I know him not," could be used, but sounds stilted nowadays.
So, where did this 'do' come from?
Constructions using the equivalent of "do" as an auxiliary for reasons of emphasis or particular meanings are known in other languages.
An example that comes to mind is Middle Welsh: In the Mabinogion, for example the phrase "oruc ... a" occurs often before another verb, and means "did and ..." or "made and ...".
It is also theorised that the "-d/t" ending of weak Germanic past tenses (eg "walked") is a remnant of an ancestor of "did", so "walked" comes originally from the collocation "walk did" (not in English but in an ancestor language).
In modern standard English we use "do" in the affirmative only for emphasis or ("I do want it"), but in some dialects, and in older forms (eg Shakespeare), it is more common. In fact in Shakespeare, we find both affirmative and negative sentences with and without "do".
I have a theory about why "do" has spread to be compulsory in the negative in modern English: I think that Jespersen's Cycle operated to create an anomalous Head-modifier construction ("I go not", where the "not" follows the head word "go"), and the modern form has spread because it is more normal in English constructions for the modifier to precede the head ("I don't go"). I've not seen this idea suggested anywhere else, though the application of Jespersen's Cycle to Old English is well established.
It comes from the Celtic languages spoken by the original inhabitants of the British Isles. (Irish Gaelic, Welsh, Cornish, etc). These languages were the original languages to have the odd use of the verb "do". In fact, there is not a single documented language on Earth outside of the Celtic languages and English to use "do" in this specific way. Here's how it ended up in English: When the Angles and Saxons (the original speakers of Old English) invaded from northern Germany, they took over most of the island of Great Britain. Because there were actually only a few thousand invaders, yet many more Celtic people there, it was not too long before most of the people speaking Old English were native speakers of those Celtic languages, and they spoke it the same way they spoke their native languages - with the odd use of "do". That's why a sentence such as "Have you a dog?" sounds old-fashioned today: it is old-fashioned. It is the original form of Old English.
Hope this helped! :)
This comment is correct that this question is answered by McWhorter, J. PhD Linguistics (Stanford), in ch. 1 of Our Magnificent Bastard Tongue (2009). The whole chapter's too long to quote; I quote one paragraph of the conclusion on p. 60.
The judgment must be the same on Celtic’s impact on English. The facts in this particular case do not lend themselves to mere parenthetical civil surmises that Welsh and Cornish “may have influenced” English grammar, with the treatment otherwise proceeding as usual, describing meaningless do and the verb-noun present drifting into existence by themselves for no reason. The facts do not indicate that the Welsh and Cornish features merely pitched in on a process that would have happened by itself anyway. If Old English had been brought to an uninhabited island—or, say, Cyprus, Greenland, or Fiji—rather than an island where Celtic languages were spoken, then there would be no such thing as a Modern English sentence like Did you see what he’s doing? That sentence would be rendered as See you what he does?, as it is in any normal Germanic—or European—language.
