What does 1 Corinthians 11:16 mean?

Paul began this section with a reference to maintaining the "traditions" he taught to the Christians in Corinth. The term used for "traditions" in that case literally refers to a teaching "passed along" or "handed down" from one person to another. That instruction can be good (2 Thessalonians 2:15), or bad (Matthew 15:2), depending on its content. Perhaps those traditions included this practice of having women keep their heads covered when participating in the services through prayer or prophesying.

The overall point being made in this passage is that culture and society interpret behaviors and styles of dress to have certain meanings. That includes what certain clothing styles, and the covering or revealing of certain body parts, "means" in that society. In the case of ancient Corinth, an uncovered woman's head was interpreted as a sign of sexual availability, prostitution, or idol worship. As such, it was inappropriate for Christian gatherings. So, too, was the opposite: a man sending mixed signals or associations by covering his head in a Christian meeting (1 Corinthians 11:4–5).

These cultural standards were—at that time—almost universal and a matter of common sense. As such, Paul points out that no Christian church of his era has the practice of allowing women to pray or prophesy with uncovered heads. He addresses those who might be inclined to argue with him. Those eager to "be contentious" about this issue are flatly told "I don't do this, and neither do other churches," suggesting the matter is decided. Any Corinthian who disagrees should not try to change it.