The bride was consecrated, set apart, and kept herself ready for the bridegroom to return, and wore a vail.

If the bride left the house while she was consecrated or betrothed, she would wear a veil so others would know she was betrothed. Rebekah wore a vail, and covered herself when she first came into the presence of Isaac.  Our sins need to be "covered" by Yeshua in order to stand in the presence of Yahweh.

And Rebekah lifted up her eyes, and when she saw Isaac, she lighted off the camel. For she had said unto the servant, What man is this that walketh in the field to meet us? And the servant had said, It is my master: therefore she took a vail, and covered herself. And the servant told Isaac all things that he had done. And Isaac brought her into his mother Sarah's tent, and took Rebekah, and she became his wife; and he loved her: and Isaac was comforted after his mother's death. Genesis 24: 64-67

The vail for Christians is Jesus.

By a new and living way, which he hath consecrated for us, through the veil, that is to say, his flesh; Hebrews 10:20

Being "consecrated through the veil" of the flesh of Yeshua, is like being covered with His blood.
But if we walk in the light, as he is in the light, we have fellowship one with another, and the blood of Yeshua Christ his Son cleanseth us from all sin. 1 John 1:7 

And from Yeshua Christ, who is the faithful witness, and the first begotten of the dead, and the prince of the kings of the earth. Unto him that loved us, and washed us from our sins in his own blood,  Revelation 1:5 

Being covered by the blood of Yeshua is also referred to in scripture as wearing a wedding garment--which contains the veil.

Come now, and let us reason together, saith the LORD: though your sins be as scarlet, they shall be as white as snow; though they be red like crimson, they shall be as wool. Isaiah 1:18 

"But when the king came in to look at the guests, he saw there a man who had no wedding garment; and he said to him, `Friend, how did you get in here without a wedding garment?' And he was speechless. Then the king said to the attendants, `Bind him hand and foot, and cast him into the outer darkness; there men will weep and gnash their teeth. For many are called, but few are chosen." Mat 22:11-14

And I said unto him, Sir, thou knowest. And he said to me, These are they which came out of great tribulation, and have washed their robes, and made them white in the blood of the Lamb. Revelation 7:14 

Let us be glad and rejoice, and give honour to him: for the marriage of the Lamb is come, and his wife hath made herself ready. And to her was granted that she should be arrayed in fine linen, clean and white: for the fine linen is the righteousness of saints. Rev. 19:7-8

The "righteousness of saints" is none other than faith and belief in Yeshua.
Even the righteousness of Yahweh which is by faith of Yeshua Christ unto all and upon all them that believe: for there is no difference: Romans 3:22 

And be found in him, not having mine own righteousness, which is of the law, but that which is through the faith of Christ, the righteousness which is of Yahweh by faith: Philippians 3:9 

But the righteousness which is of faith speaketh on this wise, Say not in thine heart, Who shall ascend into heaven? (that is, to bring Christ down from above:) Romans 10:6 

 A veil also covered the "holy of holies" in the physical temple in Jerusalem, which was destroyed when Yeshua was killed.

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. Exodus 26:33 

And the veil of the temple was rent in twain from the top to the bottom. Mark 15:38

Moses wore a vail.  That Christ becomes our vail is repeated.

And not as Moses, which put a vail over his face, that the children of Israel could not stedfastly look to the end of that which is abolished: But their minds were blinded: for until this day remaineth the same vail untaken away in the reading of the old testament; which vail is done away in Christ. But even unto this day, when Moses is read, the vail is upon their heart. Nevertheless when it shall turn to the Lord, the vail shall be taken away. Now the Lord is that Spirit: and where the Spirit of the Lord is, there is liberty. But we all, with open face beholding as in a glass the glory of the Lord, are changed into the same image from glory to glory, even as by the Spirit of the Lord. 2 Corinthians 3:13-18

Noah's ark, another picture of the temple and heaven, was also covered, not with a veil, but with pitch.
Make thee an ark of gopher wood; rooms shalt thou make in the ark, and shalt pitchit within and without with pitch. Genesis 6:14

The meanings of both words for pitch contain many suggestions of Christ's sacrificial death, which covers our sins, and was the ransom sacrifice that cleanses us. The bride of Christ is the temple of Yahweh today, not the physical temple, or the ark.

What? know ye not that your body is the temple of the Holy Ghost which is in you, which ye have of Yahweh, and ye are not your own? 1 Corinthians 6:19

I beseech you therefore, brethren, by the mercies of Yahweh, that ye present your bodies a living sacrifice, holy, acceptable unto Yahweh, which is your reasonable service. Romans 12:1

From my studies, I believe at the rapture, the sun will be darkened. It is interesting at one of the few mentions of "veil" in scripture to read:

And the sun was darkened, and the veil of the temple was rent in the midst. Luke 23:45

When is the veil removed in the marriage ceremony?  Right as they become man and wife, joined with the kiss.

As Christ is our covering, our vail, the veil of the temple was rent as Christ died.  Hebrews speaks in detail of the meaning of the vail being rent, in that now we need not a high priest to go behind the vail for us, but that Christ has become our hight priest.

The word "consecrated" in Scripture is used 14 times in the KJV.  Consecrated appears to mean to be set apart and made holy for the use of the Lord only.  Most times, the word for "consecrated" is male' {maw-lay'} (15 times).  Only once, the word for consecrated is "qodesh" which is usually translated as "holy" and also carries the meaning of "set apart".

But all the silver, and gold, and vessels of brass and iron, are consecrated qodesh unto the LORD: they shall come into the treasury of the LORD. Joshua 6:19

 The vail is a symbol of the consecration, being set apart.  A betrothed woman wore the vail to show she was not available for marriage.  This betrothal time period was also to make sure the virgin was not pregnant already.

A virgin bride was set apart, as Christians are to be apart from the world.

If the world hate you, ye know that it hated me before it hated you. John 15:18

If ye were of the world, the world would love his own: but because ye are not of the world, but I have chosen you out of the world, therefore the world hateth you. John 15:19

I have given them thy word; and the world hath hated them, because they are not of the world, even as I am not of the world. I pray not that thou shouldest take them out of the world, but that thou shouldest keep them from the evil. John 17:14-15

John 17:15 is commonly used as an "anti pre trib" verse.  Note the grammar.  Yeshua is praying for the second phrase, not praying for the first.  Why?  The second item is greater than the first, and includes it.  The tribulation is evil, and those who are raptured will be kept from it as promised in

Rev. 3:10.  It is a serious mistake of grammer to conclude that Yeshua is actively praying to keep us in the world during the tribulation--for that to be the case, the "not" would have to be placed after "shouldest" in the sentence.)

The point is that we are separated from the world, like a bride is set apart when she is betrothed and consecrated and wears the veil. Love not the world, neither the things that are in the world. If any man love the world, the love of the Father is not in him. 1 John 2:15

Therefore let us keep the feast, not with old leaven, neither with the leaven of malice and wickedness; but with the unleavened bread of sincerity and truth. I wrote unto you in an epistle not to company with fornicators: Yet not altogether with the fornicators of this world, or with the covetous, or extortioners, or with idolaters; for then must ye needs go out of the world. 1Cor 5:8-10

This is a very interesting point made here in 1 Cor 5, in that for us to be truly separate we would have to leave the world entirely because the world is so completely wicked.

The story of the ten virgins is a parable showing the importance of the bride keeping herself ready and prepared for the bridgroom.

Then shall the kingdom of heaven be likened unto ten virgins, which took their lamps, and went forth to meet the bridegroom. And five of them were wise, and five were foolish. They that were foolish took their lamps, and took no oil with them: But the wise took oil in their vessels with their lamps. While the bridegroom tarried, they all slumbered and slept. And at midnight there was a cry made, Behold, the bridegroom cometh; go ye out to meet him. Then all those virgins arose, and trimmed their lamps. And the foolish said unto the wise, Give us of your oil; for our lamps are gone out. But the wise answered, saying, Not so; lest there be not enough for us and you: but go ye rather to them that sell, and buy for yourselves. And while they went to buy, the bridegroom came; and they that were ready went in with him to the marriage: and the door was shut. Afterward came also the other virgins, saying, Lord, Lord, open to us. But he answered and said, Verily I say unto you, I know you not. Watch therefore, for ye know neither the day nor the hour wherein the Son of man cometh. Mat 25:1-13

For the Son of Man is as a man taking a far journey, who left his house, and gave authority to his servants, and to every man his work, and commanded the porter to watch. Watch ye therefore: for ye know not when the master of the house cometh, at even, or at midnight, or at the cockcrowing, or in the morning: Lest coming suddenly he find you sleeping. And what I say unto you I say unto all, Watch. Mark 13:34-37