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:

Paul is saying here that because the head of every man is Christ and the head of the woman is man, therefore in praying or prophesying the man must not have his head covered because that would dishonor Christ, and the woman must not have hers uncovered as that dishonors man and her position in God's order.

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.

People stumble over this all the time, but it seems clear to me that Paul is saying that if she will not cover her head she might as well be bald because her hair does NOT suffice as a covering. He is also implying that they will recognize that it is a shame for a woman to have her hair shorn, which makes it HIGHLY improbable that any women in the congregation cut their hair!

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.

Paul is adding reason to reason why women should cover our heads and men not cover theirs. The man is specifically the image and glory of God, but the woman the glory of man. So he is saying that the image of God should be on display, uncovered, but the glory of man should be covered in worship.

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.

This is another reason, a further explanation of God's order. Woman came from man, therefore man has the primary position as the image of God. Women are also the image of God but since they were taken out of man they primarily are the man's glory and Paul wants this recognized in the worship service.

10*For this cause ought the woman to have power on her head because of the angels.

This is another reason. "Power on the head" is usually interpreted as the wearing of a covering as a symbol of being under the authority of a husband or male leadership.

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.

Here Paul seems to be concerned to nip any abusive male domination over women in the bud by reminding us of this basic equality of the sexes.

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.

Here he's appealing to their own sense of what is right and proper, maybe meaning their Spirit-taught sense. In any case he expects them to recognize that women by nature DO have long hair, so he can't possibly be commanding it.

16*But if any man seem to be contentious, we have no such custom, neither the churches of God.

Apparently some were objecting to having women cover their heads. Sounds a lot like today.