Translation
Every man who is praying or prophesying while having head covered,
Go to footnote numbercauses disgrace to his head.
Go to footnote numberParaphrase
Every man who speaks to God or teaches people in public without having the accepted sign of respect for authority, causes disgrace to the one who is over him.
Footnotes
1
The Greek actually says, “having [something] hanging down from his head.” This was their way of saying the head was covered with a prayer covering, such as the Hebrew Talith. Because it hangs down, it does not refer to a hat or a type of turban which does not hang down (some turbans stay atop the head and some have a part that hangs down or wraps around the face).
2
Does this refer to his own head or to the one over him? Both are possible and the Bible scholars are divided on the issue. I think it is first of all a disgrace for the one who is above him in authority (Christ) and I think it can simultaneously refer to his own head, since, at that moment he is a representative of Christ.
THE GREEK WAY OR THE HEBREW WAY?
The historians and Bible scholars agree that the Jews prayed and worshipped with their heads covered in order to show reverence to God, and the Romans did the same. The Greeks however (and their neighbors, the Corinthians), showed reverence by uncovering their heads during times of worship of their gods.
Does this mean Paul chose the Greek way over the Hebrew way? No. In my opinion, he was not arguing about which way to show respect was best and that everyone needs to use the same sign of respect. Instead he was saying that when you worship God, you need to show God reverence and honor Him in the ways that are recognized in your social context. The Jews and Romans did this by covering their heads, and the Greeks by uncovering them, and both were right because they were recognizing the authority structures involved.