In 1 Corinthians 11, Paul teaches that a woman's head should be covered in the public worship of the church. Bear in mind, this is not an obscure passage from Leviticus. So we can't just write it off as "ceremonial law" (if you hold to such distinctions) that no longer applies to us as New Testament believers. What are we supposed to do with this? Is the modern church in error by forsaking this practice?
In actuality, Paul's instructions to the church at Corinth on this matter seem to be in response to a local custom in that region, rather than intended as a universal command to the Church at large. In other words, this passage would seem to be more descriptive of the situation at Corinth, than intended as prescriptive for believers of all times/places. Consider the following:
1) Christ never mentions the issue.
2) Paul never mentions the issue outside of this one occurrence in 1 Corinthians. If it was intended as a universal command, and an important practice in the corporate worship of the Church, one would think it would have come up in at least one other epistle.
3) Perhaps most importantly, such a command would seem to run contrary to the very heart of the New Testament, which stresses repeatedly that faith is a matter of the heart and worship a matter of the spirit. The New Testament clearly removes emphasis from the outward/physical. If taken as a universal command, this would seem to be a rather bizarre departure from the general thrust of the NT.
All that being said, Paul certainly does not discourage the Corinthians from taking part in this custom. And in fact, seems to advocate it. Which begs the question, "What merit does Paul see in this?" Well, the underlying principle behind the custom is actually quite biblical...
The head covering distinguishes between the role of men and women in the corporate worship of the church. And without getting too much into that (that's a whole other blog) let it suffice to say that God created men and ladies to serve separate functions. The Bible establishes male leadership, both in the family and the church. Thus, the practice of ladies wearing head coverings was an acknowledgement of God's intended design.
While our ladies may not wear head coverings to church in modern America, we too need to be faithful to demonstrate God's intended design/order for the role of men and women in the church. This is the primary application of the passage. The issue of head coverings is only secondary.