In the old testament the Hebrew for the veil of the temple or the tabernacle in the wilderness, is translated as the vail.
God instructed Moses to make the vail to divide the holy place from the most holy (or holy of holies) -
Exodus 26:31-33 (King James Version)
31. And thou shalt make a vail of blue, and purple, and scarlet, and fine twined linen of cunning work: with cherubims shall it be made:
32. And thou shalt hang it upon four pillars of shittim wood overlaid with gold: their hooks shall be of gold, upon the four sockets of silver.
33. And thou shalt hang up the vail under the taches, that thou mayest bring in thither within the vail the ark of the testimony: and the vail shall divide unto you between the holy place and the most holy.
During the daily services, the priest would take the blood from the sin offerings (or sacrifices) and sprinkle the blood on the veil of the temple or tabernacle.
Leviticus 4:13-17; 20 (King James Version)
13. And if the whole congregation of Israel sin through ignorance, and the thing be hid from the eyes of the assembly, and they have done somewhat against any of the commandments of the LORD concerning things which should not be done, and are guilty;
14. When the sin, which they have sinned against it, is known, then the congregation shall offer a young bullock for the sin, and bring him before the tabernacle of the congregation.
15. And the elders of the congregation shall lay their hands upon the head of the bullock before the LORD: and the bullock shall be killed before the LORD.
16. And the priest that is anointed shall bring of the bullock's blood to the tabernacle of the congregation:
17. And the priest shall dip his finger in some of the blood, and sprinkle it seven times before the LORD, even before the vail.
20. ...................and the priest shall make an atonement for them, and it shall be forgiven them.
The incense that burned on the altar of incense in front of the veil masked the smell of all that blood that accumulated on the viel during the year.
During the day of atonement, which happened once per year, the sanctuary was cleansed of all the sins that had been symbolically transferred to the tabernacle or temple during the daily services all during the year. The veil of the temple or tabernacle was also cleansed of the blood that had daily been sprinkled on it.
Leviticus 16:16; 29-30 (King James Version) 16. And he shall make an atonement for the holy place, because of the uncleanness of the children of Israel, and because of their transgressions in all their sins: and so shall he do for the tabernacle of the congregation, that remaineth among them in the midst of their uncleanness.
29. And this shall be a statute for ever unto you: that in the seventh month, on the tenth day of the month, ye shall afflict your souls, and do no work at all, whether it be one of your own country, or a stranger that sojourneth among you:
30. For on that day shall the priest make an atonement for you, to cleanse you, that ye may be clean from all your sins before the LORD.
In the new testament when Christ died on the cross the veil of the temple was torn in two, and the Greek is translated as the veil.
Matthew 27:50-51 (King James Version)
50. Jesus, when he had cried again with a loud voice, yielded up the ghost.
51. And, behold, the veil of the temple was rent in twain from the top to the bottom; and the earth did quake, and the rocks rent;
Christ's death or sacrifice allows us to go boldly into the most holy place before the throne of Grace, this is why the veil of the temple was torn in two. Now we can go boldly in prayer to God to ask for forgiveness of sins for ourselves. There is no need for an intermediary or priest.
Hebrews 10:12; 19; 20 (King James Version)
12. But this man, after he had offered one sacrifice for sins for ever, sat down on the right hand of God;
19. Having therefore, brethren, boldness to enter into the holiest by the blood of Jesus,
20. By a new and living way, which he hath consecrated for us, through the veil, that is to say, his flesh;
Ephesians 4:20-24 (KJV)
20. But ye have not so learned Christ;
21. If so be that ye have heard him, and have been taught by him, as the truth is in Jesus:
22. That ye put off concerning the former conversation the old man, which is corrupt according to the deceitful lusts;
23. And be renewed in the spirit of your mind;
24. And that ye put on the new man, which after God is created in righteousness and true holiness.