A friend of mine sent me this sermon, posted by Audio Sancto. The part on Catholic women's veils starts at about 9:43, but I encourage you to listen to all of it.
This the most beautiful explanation of the theological significance of the veil I've ever heard.
Here is a transcript:
We've talked about the meaning of the bridal veil before. Remember that on one level, as everyone knows and as St. Paul makes clear in 1 Corinthians, the veil is a visible sign that the woman is under the authority of a man.
These days, the idea of submission to the authority of her husband is frowned upon, to put it mildly. But it shouldn't be, once we realize that the bridal veil signifies the submission of this particular woman to the loving care of her husband. It signifies her trust, her confidence in his Christ-like leaderhsip. It signifies that she has chosen to follow him as a loving partner and companion. It also signifies that he has been specifically consecrated to handle that sacred vessel - to safely touch that ark - and that's something mysterious and beautiful.
But that doesn't explain why little girls would wear a veil, does it? It doesn't explain why professed virgins, nuns, religious sisters would wear veils, does it? Obviously, the mystical symbolism of a veil goes far, far beyond the relationship of one particular woman and one particular man. What does it mean? What sort of a mystery is presented us when we see a woman veiled before the altar?
It's a very great mystery. Like Our Lady, every Catholic woman, as a woman, is a living icon of the church. So when she veils herself here, in the presence of Our Lord, it's a visible reminder for all of the spousal relationship - the bridal relationship - between the Church and Christ.
That relationship between the Church and Christ is a very deep mystery, indeed. So whenever we see a veiled woman here, before the altar, be she six or be she sixty, it's a visible reminder for all of us of this spousal relationshp, this bridal relationship between Christ and His Church.
And because the veil also signifies the submission of the bride to the loving care of her husband, it means that the veil of a Catholic woman is also a visible reminder of the perfect submission of the church to the loving rule of Christ.
The veil is a visual sermon, it's a visual statement, it's a public proclamation before the Lord that He IS the Lord and that we love Him and that we are ready to obey him. It's a totally counter-cultural statement proclaiming obedience in the midst of a culture that is totally permeated with this attitude of "I will not serve."
That, in any age, but especially in ours, is a very great mystery indeed.