Q: Do women have to wear some sort of covering when praying or in a church service?
1 Corinthians 11:4-7, 10 (ESV)
Every man who prays or prophesies with his head covered dishonors his head, but every wife who prays or prophesies with her head uncovered dishonors her head, since it is the same as if her head were shaven. For if a wife will not cover her head, then she should cut her hair short. But since it is disgraceful for a wife to cut off her hair or shave her head, let her cover her head. For a man ought not to cover his head, since he is the image and glory of God, but woman is the glory of man. That is why a wife ought to have a symbol of authority on her head, because of the angels.
These comments are pretty strong, but the key to understanding them is found in verse 10 where Paul reveals the PURPOSE of a head covering: "That is why a wife ought to have a symbol of authority on her head..." 1 Corinthians 11:10 (ESV) You'll notice that Paul refers to a head covering as "a symbol of authority." He is referring to the authority/headship of a man over his wife as established by God.
In Paul's day and culture head coverings for women communicated a strong message of a woman's attitude of respect for the authority God had granted to her husband. This is actually still the case in some cultures, but in many places around the world it is no longer so. Women use other things to communicate their respect for God's order in marriage — which is the whole point of Paul's remarks.
Although in the Apostle Paul's culture a women's head covering carried a very strong message, here in American almost no one would view a woman with her head covered and conclude that she was expressing a biblical respect for her husband. In fact, when women are seen with some kind of "religious" head covering today most Americans assume that it is simply a symbol of a religious affiliation and nothing more.