"But I want you to understand that the head of every man is Christ, the head of a woman is her husband, and the head of Christ is God. Any man who prays or prophesies with his head covered dishonors his head, but any woman who prays or prophesies with her head unveiled dishonors her head—it is the same as if her head were shaven. For if a woman will not veil herself, then she should cut off her hair; but if it is disgraceful for a woman to be shorn or shaven, let her wear a veil.
“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. (For man was not made from woman, but woman from man. Neither was man created for woman, but woman for man.) That is why a woman ought to have a veil on her head, because of the angels…If any one is disposed to be contentious, we recognize no other practice, nor do the churches of God.” (1 Corinthians 11:3-10,16)
When women wear their veil, it is an outward sign of humility, submissiveness, and obedience to Christ.
Why at mass? Because we are going to Christ at His home and totally receive Him in the Eucharist. It is no longer a host of bread but in fact THE Body and Blood of Christ.
I hope this helps. If you are still unsure, since you aren’t Catholic maybe you have a nice hat you would like to wear. Either is appropriate. If you choose not to wear a veil or hat, no one will say anything to you. Traditionalists are always happy to see a new face.
