Segregating the sexes
The term burqa (alternative spellings burkah burkha, bourka) refers to several types of textile garments, used in various cultures for concealing some or all of the features of a female. The garments are meant to be realisations of Purdah, the practice of segregating the sexes in public life.
The garment most frequently associated with the word burqa today is the particularly severe Afghan burqa, which was required by the Taliban regime in Afghanistan. This is a loose-fitting tent-like cloak that covers the whole woman, including her face, from head to foot. Only a textile mesh is provided in front of the eyes so that the woman can have some limited visibility, while completely concealing her features from the outside.
The Afghan burqa, like other Purdha-promoting dress codes, has -- contrary to common belief -- no religious Islamic background. In the Afghan case the burqa is just part of the traditional culture of the southern Pashtuns, the ethnic majority of Afghanistan.
Ancient Persian origin
The origins of the burqa and the related (but less severe) garment chador (the required female dress in present-day Iran) can be traced back to the Achaemenian Persian Empire under Cyrus the Great in the 6th century BC. The Greek victory over the Persians temporarily saved the West from the burqa and the chador. Instead, it spread eastwards toward Afghanistan and India.
With the demise of Western (Greco-Roman) culture and the adoption of an Oriental religion (Christianity) many Oriental cultural elements found their way into Europe. The Christian Byzantine Empire adopted the Persian chador / burqa. In the Greek and Russian Orthodox world the traces of this can still be seen as the hair-covering shawls worn by many rural women in South-eastern Europe and Russia. The Muslim conquest of the Byzantine Empire in turn lead to the adoption of the burqa and chador in the Islamic world.
No support in the Scriptures
However, contrary to popular belief, there is no support anywhere in the Quran (Koran) for burqas, chardors or veils. There are only three places in the Quran where dress for women is mentioned, all of them quite innocuous:
(1) Surah 7:26: "O children of Adam, we have provided you with garments to cover your bodies, as well as for luxury. But the best garment is the garment of righteousness (REMARK: 'fear of God' in some translations). These are some of God’s signs, that they may take heed."
So just a “garment of righteousness” is recommended here.
(2) Surah 24:31: "And tell the believing women to subdue their eyes (REMARK: in the previous verse men are asked to subdue their eyes), and maintain their chastity. They shall not reveal any parts of their bodies, except that which is necessary. They shall cover their chests, and shall not relax this code in the presence of other than their husbands, their fathers, the fathers of their husbands, their sons, the sons of their husbands, their brothers, the sons of their brothers, the sons of their sisters, other women, the male servants or employees whose sexual drive has been nullified, or the children who have not reached puberty. They shall not strike their feet when they walk in order to shake and reveal certain details of their bodies. All of you shall repent to GOD, O you believers, that you may succeed."
OK, so here the Quran tells women to cover their breasts, hardly a spectacularly puritan advice.
(3) Surah 33:59: "O prophet, tell your wives, your daughters, and the wives of the believers that they shall lengthen their garments. Thus, they will be recognized and avoid being insulted. God is Forgiver, Most Merciful."
Lengthen, OK. But nothing is said from what starting-point. So in essence the Quran would allow the wearer of a bikini to lengthen this garment into a pair of hotpants and still be a righteous Muslim.
In summary, this is as far as the Quran goes when prescribing a dress code for women:
(1) The best garment is righteousness
(2) Cover your breasts
(3) Lengthen your garment
The backside of the polyhedron
This is of course just one side of the monotheistic polyhedron. Even if the Quran and the Bible are said to represent the very essence of their respective creeds, this has never prevented clerics to interpret, elaborate and invent thousands of completely unrelated ideas and prescriptions. The Muslim Sharia (which is NOT the Quran) is one example, the Catholic saint cult another. And every minute umpteen fundamentalist Protestant fire-and-brimstone preachers invent umpteen new interpretations of the Bible.
So even if burqas and severe dress codes for women lack support in the Quran, the burqa, the chador, the veil, and the shawl (hijab) are firmly entrenched in many Muslim denominations, not because of the Quran, but because of the imaginative interpretations and ethnical traditionalism of the clergy.
The Quran (English, German, and Swedish translations)