I had never been in a church that teaches that women are to cover our heads, so it was surprising to discover that there are such churches, and surprising also to come to the conclusion myself from my study that it's certainly what that passage requires of us.
Some churches I'd attended simply taught nothing at all on the subject, though various women in different churches understood the required covering to be long hair, basing this on verse 15 of the passage where Paul speaks of long hair as a natural covering to women. (I've reproduced the passage below so you can check this out there).
My church at the time subscribed to the position of the Council on Biblical Manhood and Womanhood, http://www.cbmw.org/ which holds that the covering required by Paul is to be understood as merely the cultural symbol in Paul's time of female submission to male headship in the church, and since it doesn't have that meaning today we should substitute whatever conveys that meaning in our time. (However, one thing I found out in my study was that it was NOT the cultural symbol of submission they claim it was, and besides seeing this in pictures on Google Image of pre-Christian Greek and Roman male and female dress, all the proof you really need is the fact that Paul is exhorting them at such length to do this. A secondary point is that there is absolutely NOTHING that conveys that symbolic meaning in our time anyway. We have to learn from Paul that covering the head has that meaning TO GOD AND TO THE ANGELS.) Check out Questions 31 and 32 on this page: http://www.cbmw.org/Online-Books/Fifty-Crucial-Questions/Fifty-Crucial-Questions#31
Nevertheless, interestingly, at their website they also offer for online reading a book by Mary Kassian, Women, Creation and the Fall, in which she has a chapter about the meaning of this passage and concludes that it does refer to covering the head and that it applies to us today. http://www.cbmw.org/images/onlinebooks/women_creation_fall.pdf Go to Chapter 9, page 92, about halfway down the PDF document. You can skip to the end of the chapter to find her conclusion that the head covering is still applicable today and that she would "encourage" women to cover their heads in church.
OK, I found I could copy out the quote. Here it is, from page 104 at the very end of the chapter:
In 1 Corinthians 11, Paul exhorts the Corinthian church to adhere to a custom he had taught them: Women were to veil themselves in the public assembly of believers, and men were not. This was a distinctly Christian custom observed out of respect for the principle of headship. The reasons Paul gives for this custom are cross-cultural. They appeal to creation and the laws of nature. This reveals to us how intricately the principle of headship is woven into the created order. Whether one believes that women should veil themselves today (as I would encourage) or that headship could be affirmed by other appropriate forms of dress in our day, the conclusion is the same: The headship principle is so fundamental, so central to the functioning of the local church that it must be be evidenced symbolically in the Christian meeting. Headship is the same principle which underlies marital structure. And this vital doctrine lays foundation for the balance of New Testament teaching on the role of women in the church.
The reason it took me so much time and effort to get to what now seems like a pretty straightforward reading of the passage, is the variety of interpretations held by different branches of the church, and the vehemence with which they are held.
Just recently I discovered a preacher quite new to me whom I appreciate very much, Zac Poonen of Bangalore, India. I'm a bit of a heretic to some branches of the church, particularly the Reformed, in that I'm attracted to the "higher life" teaching, which is often dismissed by them as "mysticism," a word that is badly misused by them in my opinion. That's a whole discussion unto itself, but I mention it because that is Zac Poonen's attraction for me also. In exploring his sermons because of this interest in his basic message, I discovered that he happens to have a teaching on the passage about the head covering and was very happy to find that he agrees with me! And says it wonderfully succinctly: http://downloads2.sermonindex.us/16/SID16538.mp3 (There is no counter on this audio device and he covers the whole chapter, also spending about six minutes at the beginning on the first verse which doesn't apply to this topic, so to get just the part about the head covering see if you can set the pointer about a sixth to a fifth of the way along the track. The message runs about 18 minutes.)
1 Corinthians 11:
2 Now I praise you, brethren, that ye remember me in all things, and keep the ordinances, 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, having his head covered, dishonoureth his head.
5 But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head: for that is even all one as if she were shaven.
6 For if the woman be not covered, 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, 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 on her head because of 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.
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?
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, we have no such custom, neither the churches of God.