Worship and Communion in the Early Church (1Corinthians 11)

2. The Argument from Order of Creation (11:7-12)
Now Paul moves to a second argument, the argument from the order of creation.
"7 A man ought not to cover his head, since he is the image and glory of God; but the woman is the glory of man. 8 For man did not come from woman, but woman from man; 9 neither was man created for woman, but woman for man. 10 For this reason, and because of the angels, the woman ought to have a sign of authority on her head. 11 In the Lord, however, woman is not independent of man, nor is man independent of woman. 12 For as woman came from man, so also man is born of woman. But everything comes from God." (11:7-12)
Woman is the glory of man, Paul is saying, because according Genesis 2:21-23, woman made from man's rib. Apparently, in this argument, the woman's head should be covered to hide the "glory of man" before God. But so as not to reinforce male dominance, he notes that males and females are interdependent (11:11-12). Again, this argument probably had more impact in the first century than it does today.
Authority on Her Head (11:10)
Perhaps the key verse to understand here -- and the most mysterious -- is verse 10:
 "For this reason, and because of the angels, the woman ought to have a [sign of] authority on her head."
Probably the words "authority" and "angels" were used in a letter of inquiry sent to Paul by the Corinthians, but without that context, we don't know exactly what Paul intended.
"Because of the angels" probably refers to Paul's belief that angels, as witnesses to the order of creation, attend worship services, and are "concerned that the ordering of things established at the creation is maintained," thus supporting Paul's argument from creation in verses 7-12. Two Qumran parallels seem to support this view.
One important question is what it means for the woman to "have authority" on her head. The words "sign of" (NIV) or "symbol of" (NRSV) do not appear in the text and are an interpretation, not a translation. The noun is exousia, "authority," probably referring to "a state of control over something, freedom of choice, right (e.g., the 'right' to act). " What does it refer to here? Several interpretations have been suggested.
"The veil is not a sign of a woman's submission to her husband's authority, nor even of her social dignity and immunity from molestation; it is a sign of her authority.... In Christ she had received equality of status with man: she might pray or prophesy at meetings of the church, and her veil was the sign of this new authority."

1. Man's authority over the woman. This requires us (a) to take exousia, "authority" in a passive sense (which doesn't occur elsewhere in the New Testament); and to take the idiom "to have authority over" as an external authority different from the subject of the sentence, which likewise does not occur elsewhere.389 According to Jewish custom, a bride went bareheaded until her marriage, as a symbol of her freedom; when married she wore a veil as a sign she was under the authority of her husband. According to this interpretation, as confidently stated by the Living Bible, "So a woman should wear a covering on her head as a sign that she is under man's authority."
2. Sign or means of exercising authority to pray and prophesy. Thus, the authority to pray and prophecy in public, previously denied to women in the synagogue, was now symbolized by the woman's head covering. This is attractive but speculative, since there isn't enough evidence in the text to support it. F.F. Bruce says,
3. Authority over her head to do as she wishes. This takes exousia in its normal sense and the preposition epi as "over," but it seems to contradict Paul's argument here.

Ultimately, what Paul meant here is rather obscure, which is problematic when people attempt to build doctrine on the basis of this verse.
The so-called "Covering Doctrine," popular in non-denominational Charismatic groups and elsewhere, uses the word "covering" to refer to which authority a person is submitted to. It contends that all people in the church need to be properly "covered" by God's "delegated authority" in the church. As you can see, it is a big stretch to use verse 10 to support anything like this doctrine. Certainly, we need to respect the spiritual authorities that God has put in place, but not in a servile manner.