Should women use head coverings when praying?
Consider also this similar question: Is it proper for a man to pray while his head is covered?
The overall point in I Corinthians 11:1-16 is a discussion concerning authority, leadership, and submission. Paul states, "I want you to know that the head of every man is Christ, the head of woman is man, and the head of Christ is God" (I Corinthians 11:3). The showing of submission to those over us is done in a variety of ways.
Submission can be shown in our willingness to obey those who have authority over us. For example, slaves were required to obey their masters. "Bondservants, obey in all things your masters according to the flesh, not with eyeservice, as men-pleasers, but in sincerity of heart, fearing God" (Colossians 3:22). The reason was simple, our actions reflect on the one we call our Master, God the Father. "Let as many bondservants as are under the yoke count their own masters worthy of all honor, so that the name of God and His doctrine may not be blasphemed" (I Timothy 6:1).
Giving honor to those over us also shows a willingness to submit. "For in this manner, in former times, the holy women who trusted in God also adorned themselves, being submissive to their own husbands, as Sarah obeyed Abraham, calling him lord, whose daughters you are if you do good and are not afraid with any terror" (I Peter 3:5-6). Sarah demonstrated her willingness to follow Abraham when she addressed him by the title of lord. That is not to say all wives should call their husbands "lord," but it is an illustration of the extent of Sarah's respect for her husband. The same goes for our other relationships. "Honor all people. Love the brotherhood. Fear God. Honor the king" (I Peter 2:17).
Submission is also shown in our interactions with others. Gentleness and quiet manners show our willingness to serve. "For to this you were called, because Christ also suffered for us, leaving us an example, that you should follow His steps: "Who committed no sin, Nor was deceit found in His mouth"; who, when He was reviled, did not revile in return; when He suffered, He did not threaten, but committed Himself to Him who judges righteously" (I Peter 2:21-23). The same applies to our relationship with the government. "Remind them to be subject to rulers and authorities, to obey, to be ready for every good work, to speak evil of no one, to be peaceable, gentle, showing all humility to all men" (Titus 3:1-2).
Finally, submission is demonstrated by what we choose to wear. In Paul's discussion of submission to Timothy, he stated, "in like manner also, that the women adorn themselves in modest apparel, with propriety and moderation, not with braided hair or gold or pearls or costly clothing, but, which is proper for women professing godliness, with good works" (I Timothy 2:9-10). If you saw a young man sporting a mohawk haircut, body piercings, and tatoos and compared him with another young man wearing a trim business suit, whom would you conclude is rebellious and whom conforming? Our choice of attire is an outward reflection of our inward attitudes.
In I Corinthians 11, Paul talks about dress in regards to praying and prophesying. He states that a man wearing anything on his head while praying or prophesying brings disgrace on his head, that is Christ (I Corinthians 11:4,7). For women, Paul states that a woman praying or prophesying with her head uncovered dishonors her head, that is man (I Corinthians 11:5-6, 13). To prove that God's requirement is not outrageous, Paul points out that men naturally favor short hair and women favor long hair. "Does not even nature itself teach you that if a man has long hair, it is a dishonor to him? But if a woman has long hair, it is a glory to her; for her hair is given to her for a covering" (I Corinthians 11:14-15).
Therefore, Paul is arguing for women to wear a head covering while praying and prophesying. See the references below for further details.