Some women cover their heads at Mass. Here is what it is and is not.
A veil, a mantilla, a scarf, or a hat. Any of them count, and the material has never been the point.
It goes on entering the church and comes off leaving it. Some women veil only in the presence of the Blessed Sacrament, including adoration.
If you want to try it, sit where you are comfortable and expect to feel conspicuous for about two weeks. That passes.
The traditional reading is that in Catholic practice sacred things are veiled: the tabernacle, the chalice, the sanctuary in Lent. A woman who veils is claiming that reading for herself, as a sign of what carries life. Others veil more simply as a mark of prayer and modesty before the Blessed Sacrament.
Saint Paul writes to Corinth about women praying with their heads covered, and head covering was a normal part of Western Catholic practice for most of the Church's history, obliged in the 1917 Code of Canon Law.
The 1983 Code, which replaced it, simply does not carry the requirement forward. Nothing was condemned. The law was not renewed, so the obligation ended and the practice became free.
It has returned in many places over the last two decades, chosen rather than assumed, which is a genuinely different thing from what it was in 1950.
No woman is obliged to veil. It is not a measure of piety, it is not required for reverence, and it is not a judgment of women who do not. Anyone who uses it as a way to rank other people in the pews has already lost the meaning of it.