Question:
With regards to 1 Corinthians 11:5-6, and 1 Corinthians 11:13-15, is it necessary for women to cover their head when they pray, or is it enough to have long hair?

Answer:
1 Corinthians 11:2-16 was written by Paul because of a very special circumstance in Corinth. In Corinth, the prostitutes wandered outside of the temple with shaved heads. Their shaved heads were not only an advertisement of who they were, but also a sign of their rebellion to God's order. In response to that, Paul wrote the following three sections:

Section #1
1 Corinthians 11:2-3: The Tradition
Verses 2 and 3 describe the tradition handed down by Paul to all churches (see verse 16 below) regarding God's order of things; specifically that the head of the man is Christ, the head of the woman is man, and the head of Christ is God.

Section #2
1 Corinthians 11:4-12: The Explanation
Beginning with verse 4, Paul explains how this tradition applies directly to the church in Corinth. Verse 4 states that for a man to have his head covered during prayer or prophecy is a shame, while verse 5 says that it is a shame for a woman to have her head uncovered during prayer or prophecy.

Verse 6 gives two commands: Command #1 If a woman wants no covering for her head, let her be shaved (since she will be rebellious to God's order). Command #2 If it is a shame to her to have shorn or shaved hair, let her be covered (if she desires to show God's order).

Verse 7 explains that it is necessary for the man to be uncovered because he is made in the image and glory of God, but the woman is the glory of man.

Verses 8 and 9 gives the reasons for Verse 7. Reason #1 - Man is not from out of woman, but the woman is from out of the man. Reason #2 - The man was not created for woman, but the woman was created for the man.

Verse 10 says that because of these two reasons, it is necessary that the woman have a covering on her head, because of the angels. Angels watch the church to see God's order. Paul is saying that the angels should not see rebellion to God's order in the church.

Verse 11 states that in spite of the stated truths, man is not to separate from (independent) the woman, and the woman is not to separate (independent) from the man. Verse 12 gives the reason for this - woman was made from out of man, and man was made through the woman and everything is from God.

Section #3
1 Corinthians 11:13-16: The Instruction
Verses 13 and 14 ask two questions. Verse 13, asks, Is it proper for a woman to pray to God uncovered? Verse 14, asks, Does not nature itself teach that if a man has long hair it is a dishonor to him?

Verse 15 makes the statement that if a woman should have long hair, it has been given to her as a covering.

Verse 16 says there is no other teaching, regarding the Lord's order, in the church.

Conclusion
Paul taught God's order in every church. In the city of Corinth, the women who were rebellious to God's order and were involved in immorality shaved their heads. Therefore Paul commanded the women in the church in Corinth to choose what they wanted their lives to express - if rebellion, then they were to shave their heads; if submission to God's order, then they were to let their hair grow long. All of the words about "the covering" in verses 2-14 are said to be a woman's hair in verse 15. This was a special incident in Corinth and no other churches were commanded to do this.