As a Christian leader who is a woman, I am occasionally asked by Muslim friends whether Christian women should cover their heads as many Muslim women do. For some Christians in other parts of the world, it is customary for women to cover their heads. But for women in North America, it is a very small minority who cover.
I want to begin by saying that, as far as I know, the only place in the New Testament that mentions this is in one of the apostle Paul’s letters to the church at Corinth (First Corinthians chapter 11). In chapters 11-14, Paul is addressing several problems concerning the worship practices of the Corinthian community, and the discussion of head coverings in chapter 11 is apparently a response to something Paul has heard about what’s going on in Corinth.
As biblical scholar Richard Hays reminds us, when we read Paul’s letters, we are literally reading someone else’s mail. The letter was originally addressed to a young mission church to address particular issues.
In these verses in chapter 11, Paul endorses the freedom of women to pray and prophesy in the assembly, although there is the question about what sort of head covering is appropriate for them while exercising this freedom. The patriarchal order of verses 3 and 7-9 seems to be in conflict with a vision of mutual interdependence of men and women “in the Lord” (verses 11-12). The passage does not require subordination of women, even though some of Paul’s arguments assume a hierarchical ordering, a symbolic distinction between the sexes. The immediate concern of the passage is for the Corinthians to avoid bringing shame on the community.
In reading other parts of First Corinthians it seems that some of the church members had enthusiastically embraced the early Christian tradition that “in Christ there is neither slave nor free, there is no male and female, for you are all one in Christ.” (Galatians 3:28) In keeping with this belief, at least some Corinthian women were removing their head coverings or letting their hair down in worship, consciously discarding a traditional marker of gender distinction. Some scholars believe that Corinthian women who rejected head coverings were expressing their transformed spiritual status “in Christ.”
The Greek word for “veil” does not appear anywhere in this passage. Some biblical scholars have suggested that the whole passage deals not with wearing a veil, but with having the hair bound or unbound. In a more literal translation, we might understand that to have the head “covered” would mean to have the hair tied up on top of the head, rather than hanging loose. So there is some ambiguity here about what they were talking about.
In interpreting the Bible, you could take one verse out of context to prove a point you might want to make. People do it all the time. For example, someone might quote 1 Corinthians 7:1, “It is good for a man not to have sexual relations with a woman,” or to quote the apostle Paul to those who are unmarried, “It is good for them to stay unmarried, as I do,” without reading the rest of the passage for the full meanings. But these would not be faithful interpretations of the scripture.
However, if you read the whole section of chapter 11, you find that Paul is arguing to get the women back in line with some of his earlier teaching and with the customary practice he has fostered in his other churches. It seems clear that Paul hopes the women in Corinth will come over to his understanding of the need for head coverings in the worship setting. But then, midway through 11:2-16, Paul seems to step back and question the implication of his one-sided argument for women to wear head coverings.
In his eagerness to bring the Corinthians women into conformity with the practice of women in his other churches, Paul has subordinated them to men and husbands. Earlier in this letter, he has already taken a counter-cultural position in how he describes the relationship of husbands and wives in marriage (chapter 7) as one of equality and mutuality. It seems that Paul recognizes that two of his own values are in conflict. On one hand, Paul wants women believers to accommodate to cultural practices of wearing a head covering in worship. On the other hand, he believes that in Christ the cultural differentiations between men and women are being challenged by the gospel. 
Recognizing that he has presented the Corinthians with conflicting arguments, he leaves it to their own discernment, even though he clearly prefers that they conform to the custom of women covering their heads: “Weigh out these matters among yourselves.” “Judge for yourselves.” (11:13)
That is what we do. We study the relevant passage(s) and the context and the fullness of our scriptures. I consider how many times Jesus talked about his commandment to love God and neighbor and about our call to live more fully into what he described as “the kingdom of heaven”. In the big picture, there is one passage in which the apostle Paul advocates for women covering their head (whatever that meant to the early church in Corinth) and then leaves it up to them to decide.
I respect the decision of other women to choose to cover their heads (or not), and I cover my head when I visit a mosque, out of respect for that tradition. But I expect the same respect for my decision.