"Every man who prays or prophesies with his head covered dishonors his head." NIV translation
Why do Jews and many religions require the man's head to be covered while, here, Paul says it dishonors his head? The rabbis of the day had taught that man should cover his head based on the fact that Moses covered his head when he came down from receiving the Ten Commandments. They misinterpreted Moses' actions as he was covering his head so that the people would not see that the glory of the Lord was fading from him.
"A man ought not to cover his head, since he is the image and glory of God; but the woman is the glory of man. For man did not come from woman, but woman from man; neither was man created for woman, but woman for man. For this reason, and because of the angels, the woman ought to have a sign of authority on her head." NIV translation
Paul recounts the creation of man and woman in the garden. He goes on to say that, as an example to the angels, women are to demonstrate their subjection to Christ through the covering of their head in worship.
"In the Lord, however, woman is not independent of man, nor is man independent of woman. For as woman came from man, so also man is born of woman. But everything comes from God. Judge for yourselves: Is it proper for a woman to pray to God with her head uncovered?" NIV translation
He goes on to explain that Jesus changed everything and that through Him comes freedom. A man or a woman can worship God whether their head is covered or whether it is not. The heart is what matters and we are free to listen to the Spirit in all things and that includes the covering of the head.