EDITED Nov 12, 2012
Let's read and study Pauls instructions in 1 Cor. 11:
1 Be ye followers of ME, even as I also am of Christ.
2 Now I praise you, brethren, that ye remember ME in all things, and keep the ordinances (CUSTOMS of that time and location, NOT GODS COMMANDMENT), as I delivered them to you.
3 But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God.
4 Every man praying or prophesying (works of the ministry), having his head covered, dishonoureth his head.
5 But every woman that prayeth or prophesieth (works of the ministry) with her head uncovered dishonoureth her head: for that is even all one as if she were shaven (but she's not shaven).
6 For if the woman be not covered (with a head covering, NOT talking about hair), let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered.
7 For a man indeed ought not to cover his head (with a head covering while doing the works of the ministry in services), forasmuch as he is the image and glory of God: but the woman is the glory of the man.
8 For the man is not of the woman: but the woman of the man.
9 Neither was the man created for the woman; but the woman for the man.
10 For this cause ought the woman to have power (sign of authority under the man) on her head because of the angels (because they are intensely interested in the relationship of GOD and man and we will eventually judge the angels).
11 Nevertheless neither is the man without the woman, neither the woman without the man, in the Lord.
12 For as the woman is of the man, even so is the man also by the woman; but all things of God. (don't over control the woman)
13 Judge in yourselves: is it comely that a woman pray unto God uncovered?
14 Doth not even nature itself teach you, that, if a man have long hair, it is a shame unto him? (great example and only an example to be learned from nature. He is NOT saying that he was talking about hair the whole time in verses 3-13) (also note that a man CAN have long hair in the exception of a Nazarite of Numbers 6 as was Samson, Paul himself and Christ himself)
15 But if a woman have long hair, it is a glory to her: for her hair is given her for a covering.
16 But if any man seem to be contentious (in dispute), we have no such CUSTOM, neither the churches of God. (women do NOT have to wear head coverings in services if it will cause a dispute or discord or problem or division! This is the principal of Romans 14, not causing people to stumble by your worship.)
I love this note in the KVJ Zondervan Study Bible:
"Paul is not correcting belligerent Christian women who are coming to the assembly with their heads uncovered, for he praises them for doing what they have been taught (v.2). Rather, he wants them to understand why (v.3) it is that when praying or prophesying men's heads are to be uncovered and women's covered. Due to God's creation plan for man and woman as evidenced in the creation order of Gen 1-2, the distinctions in male and female roles need to be observed at those times during which GOD allows women to perform seemingly male roles of leadership and teaching (see 1 Tim 2:12). So when leading in public prayer and when exercising the spiritual gift of prophesying, women are to demonstrate the authority which is over them. ...."
There are no OTHER biblical instructions concerning head coverings. Paul said twice that this was the custom (of his time) that HE taught. He never said it was the commandment of GOD. So, In conclusion, women do NOT need to wear headcoverings today.