"What exactly is a "veil"? Is it a scarf that you put on your head? Why is this done?"
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This is a good source for chapel veils, mantillas, etc. Those things including hats qualify as head coverings for Church. http://www.halo-works.com/shopping/veils.html You can browse through the site and get a sense for what we're talking about.
The Apologia site is an excellent resource on all things traditional. http://www.kensmen.com/catholic/index.html
On this page she gives a very good explanation as to why we wear veils. I'll excerpt some of it below.
http://www.kensmen.com/catholic/theveil.html
You will recognize this, GipperGal, from the historical Maronite church architecture. The "curtain" or veil was traditional for the Syrian Aramaic Church. The first Maronite Churches were constructed on top of Jewish synagogues and to this day, retain some of their original elements.
The Ark of the Tabernacle was a wooden casket used as a repository for the holy things (Tablets of the Law). It was considered the throne or dwelling place for the Lord. The Ark in the Holy of Holies was protected and concealed by a veil to emphasize the "shekinah" or glory of God under His tent.
It is significant to note the development in the Christian understanding of the temple which was identified with the Body of Christ. The Church was not merely the symbolic but the real presence of God. This new temple was made up of the apostles and prophets who were the foundations of the temple. Christ was its cornerstone and principle of cohesion (unity) and each Christian was a living stone set on the foundation by the Holy Spirit. Together with Christ, the community of believers made up His dwelling place. (1Cor. 3,9; 2 Cor. 6, 16; Ephs. 2, 19-22). This earthly temple was a forshadowing of the heavenly temple (Apoc. 3, 12; 7, 15).
It is this understanding handed on by the early Church, which formed the attitudes for Christian worship and formulated the guidelines for Church architecture in the Syro-Antiochene Church, the oldest tradition in the Catholic Church.
That tradition is retained in the Maronite Divine Liturgy, with the Prayer of the Veil.
O Lord,
may your peace, true love,
and eternal and divine grace
be with us and among us all the days of our lives,
and we will give you glory and thanksgiving,
now and forever.
Some of the Maronite Churches are now restoring the curtain .... veil. Vatican Council II Fathers challenged the Church to interiorize its faith and renew its public worship. This need to return to the sources is due to the fact that over the years, the Maronite Church, wishing to manifest its unwavering fidelity to the Church of rome, readily accepted Roman adaptations and latinizations. As a result, the Maronite Church has fallen away from its original traditions. The deep thirst to rediscover the original Maronite traditions has been taken up by each diocese and by the Church Synod.