Throughout the ages Catholic women have always covered their heads while at church. This part of our identity fell widely out of practice in the late 1960's.
But now it appears that headcovering, and in particular the chapel veil is making a comeback. I for one am grateful to be part of its return!
Some people ask, "Why cover your head? Isn't that antiquated? Isn't that out of date?"
To answer, I can only explain this way: veiling is an outward sign. Our Catholic faith is full of external signs and expressions that point us to the supernatural; they point to the mysteries of our faith. When we let these precious customs fall by the wayside, we lose something beautiful. Maybe we lose some part of that which has the potential to turn our mind and heart to God. Can the chapel veil help to turn a woman's mind and heart to God? I think so.
Maybe back in the old days many women covered because, well, that's just what everyone did... St. Paul wrote that 'women must have a covering over their heads', so that's the way it always was. Then the 1917 Code of Canon Law Church required it. And frankly that's reason enough--as far as I'm concerned. That's simple enough.
In the past, in covering, it was also understood that the woman submits to the authority of her husband, which is a sore point for many today, certainly. Well what can be said about that? If the headship of the husband is understood in light of the fact that the husband is likewise under God's authority, and that he has no right to ask anything of his wife that would go against God's laws, then I believe that the issue of authority gets set in the proper perspective.
Furthermore, as St. Paul affirms, a man must love his wife as Christ loves the Church and gave himself up for her. That's a very tall order. Perhaps women have the better deal within the relationship, because it's much easier to submit than to love as Christ loves.
The 1983 Code of Canon Law said nothing regarding the issue of headcovering. So now some argue that it was never officially abrogated, since Canon Law states that when in doubt, the older law stands. Still most others say it was abrogated...
In any event, in most places nowadays, a woman has a choice about whether to veil or not. In most places, no one's going to lead a woman out into the vestibule and hand her a chapel veil to borrow. So it's a choice.
For many of us that choose to wear veils--now, in our time-- I think it's motivated by the desire for an affirmation of what we believe in our hearts: that Our Lord Jesus is truly present in the Most Holy Sacrament on the altar. That's the sense I get from those with whom I've spoken, anyway. And yes, I know many Catholic women believe this without wearing a veil. Of course, they believe but don't choose to go without that outward sign.
In what way is the veil an outward sign? 1 Corinthians: 11 explains that a woman's hair is her glory.
1 Cor 11: Doth not even nature itself teach you, that a man indeed, if he nourish his hair, it is a shame unto him?  But if a woman nourish her hair, it is a glory to her; for her hair is given to her for a covering.
So perhaps the veil is an outward sign in this way: A woman covers her hair--which is a sign of her glory--in the presence of One who is far more glorious: Our Lord and Saviour Jesus Christ in the Most Blessed Sacrament. In doing so she submits to Him.
This is what I've come to believe about the chapel veil. That's what it's about for me. Our Lord Jesus is with us: body, blood, soul and divinity in the Eucharist. How wonderful is that mystery of our faith? What a privilege it is for women to have some external means of testifying to that; of showing some outward sign of reverence to Jesus in the Eucharist!
So, we cover our head as a way of acknowledging where we are, and genuflect before Our Lord in the tabernacle as a sign, as an expression of love for Our Lord Jesus.