The veil that is mentioned in the Old Testament was a veil that God commanded Moses to created between the holy place and the Holy of Holies where a priest entered only one time a year and that with a sacrifice. This is mentioned in the Old Testament and also in Hebrews 9:2-10. The priest would offer up a sacrifice for himself and for the people before entering the Holy of Holies. This place in the center of the temple represented where God dwelt.
It is interesting that when Jesus died on the cross, it says in Matthew 27:51 that this veil was torn. It was torn from top to bottom. From God to men because of what Jesus did. The way to God was open now to anyone at any time because of what Jesus did. I wonder to myself what those Pharisees and Sadducees that saw the veil torn would have made of it. I wonder to myself if they tried to sew it back together. It seems to me that in a sense man is still in many ways trying to sew it back together, so that we can come to God on our own terms instead of his.
In his little book "The Pursuit of God" by A.W. Tozer, the author devotes a chapter titled "Removing the Veil" to this subject. Tozer comments that what we have today is not a veil of fabric, but a veil over the heart. For the christian, this creates distance in his or her relationship with God. Tozer identifes some of the threads of this veil as self-sins of "self-righteousness, self-pity, self-confidence, self-sufficiency, self-admiration, self-love". This veil is strong in the church today and in our world. In fact, most of those qualities are applauded in our world today.
Tozer goes on to say in his book, which I would recommend to you by the way, that our part in removing the veil is to pray and yield and surrender this to God and allow him to remove it. The veil is torn not from bottom to top, but from top to bottom. Tozer goes on to say, "We must confess, forsake, repudiate the self-life, and then reckon it crucified." (Tozer, p.47). As a believer, I must be careful because this world has a firm hold on those things that Jesus told us to rid ourselves of. The world applauds pridefulness and self-centeredness and those who accumulate toys and things. Rather, my desire is to be rich towards God and not to be seen sewing the veil back together.