The veiling of women was not an Islamic innovation but a Persian and Byzantine-Christian custom that Islam adopted. For most of Islam’s history, the veil in its various forms was seen as a sign of distinction and protection for upper-class women. Since the 19th century, the veil has come to represent a more assertive, self-consciously Islamic expression, sometimes in reaction to Western currents--colonialism, modernism, feminism.
The Veil in the Quran
Initially in Prophet Muhammad's life, the veil was not an issue. His wives didn't wear it, nor did he require that other women wear it. As he became more important in his community, and as his wives gained stature, Muhammad began adapting Persian and Byzantine customs. The veil was among those.
The Quran does address veiling explicitly, but only in so far as the Prophet’s wives were concerned. The wives were to be "covered," that is, unseen, when in company of other people. Significantly, the Quran's requirement did not mention a veil as it’s understood in the West—as a face covering—but a hijab, in the sense of a "curtain," or a separation of sorts. Here is the relevant passage in the Quran, best known as the "Verses of the Curtain":
Believers, do not enter the houses of the Prophet for a meal without waiting for the proper time, unless you are given leave. But if you are invited, enter; and when you have eaten, disperse. Do not engage in familiar talk, for this would annoy the Prophet and he would be ashamed to bid you go; but of the truth God is not ashamed. If you ask his wives for anything, speak to them from behind a curtain. This is more chaste for your hearts and their hearts. (Sura 33:53, N.J. Dawood translation).
What Led Muhammad to Require Some Covering
The historical context of that passage in the Quran is instructive. Muhammad's wives had been insulted on some occasions by members of the community, leading Muhammad to see some form of segregation for his wives as a protective measure. One of Muhammad's closest companions, Omar, famously chauvinistic, pressured Muhammad to limit women's roles in his life and to segregate them. The Verses of the Curtain may have been a response to Omar’s pressure. But the event closest connected to the Quran's Verses of the Curtains was Muhammad's wedding to one of his wives, Zaynab, when guests wouldn't leave and acted improperly. Shortly after that wedding, Muhammad produced the "revelation" of the curtain.
Regarding manners of dress, and other than that passage, the Quran requires only that women and men dress modestly. Beyond that, it never requires face or full-body coverings of any form for men or women.