What does it mean to be the Bride of Christ?
Being the Bride of Christ means the Church is formed entirely from Christ, is united to Him as bone of His bone and flesh of His flesh, and will one day be presented to Him pure and glorified at the Marriage Supper of the Lamb.
To be the Bride of Christ means the Church has no origin apart from Christ, the second Adam. Just as Eve was formed only from Adam's rib with no foreign substance added, the Church, the Bride of Christ, is formed entirely of Christ, out of His side, by His blood. Ephesians 5:30 says we are members of His body, of His flesh, and of His bones. There is no other source for the Church's existence; everything it takes to make the Bride comes from Christ Himself.
This bride relationship also carries the pattern of marriage. As Ephesians 5:22 through 32 explains, the husband is head of the wife even as Christ is head of the Church, and the Church is to be subject to Christ alone, not to any man. Christ loved the Church and gave Himself for it, that He might sanctify and cleanse it with the washing of water by the word, and present it to Himself a glorious Church, not having spot or wrinkle, but holy and without blemish. This is called a great mystery, for it is Christ and the Church being spoken of through the picture of a man leaving his father and mother to be joined to his wife as one flesh.
Being the Bride also means sharing in the same power and dominion given at creation. Just as Eve had the same dominion and power as Adam even while her flesh was in submission to him, the Church, as the Bride, has been given the same works and power that Christ has, for He said the works that He does, His people will do also. Submission to Christ as head does not diminish the Bride's power to operate in the things of God; it simply means the Bride belongs wholly to Him and honors Him as her husband.
Finally, being the Bride of Christ points forward to the Marriage Supper of the Lamb, the great celebration in Heaven that comes after the vial judgments, when the redeemed Church, purified and glorified, is united with Christ in perfect fellowship. Revelation 19 describes the Bride as having made herself ready, arrayed in fine linen, clean and white, which is the righteousness of saints, and all who are called to that marriage supper are declared blessed. This marks the beginning of everlasting communion with Christ, when the Church enters eternal fellowship with Him and prepares to reign with Him in glory forever.
The scriptures
Wives, submit yourselves unto your own husbands, as unto the Lord. For the husband is the head of the wife, even as Christ is the head of the church: and he is the saviour of the body. Therefore as the church is subject unto Christ, so let the wives be to their own husbands in every thing. Husbands, love your wives, even as Christ also loved the church, and gave himself for it; That he might sanctify and cleanse it with the washing of water by the word, That he might present it to himself a glorious church, not having spot, or wrinkle, or any such thing; but that it should be holy and without blemish. So ought men to love their wives as their own bodies. He that loveth his wife loveth himself. For no man ever yet hated his own flesh; but nourisheth and cherisheth it, even as the Lord the church: For we are members of his body, of his flesh, and of his bones. For this cause shall a man leave his father and mother, and shall be joined unto his wife, and they two shall be one flesh. This is a great mystery: but I speak concerning Christ and the church.
And I heard as it were the voice of a great multitude, and as the voice of many waters, and as the voice of mighty thunderings, saying, Alleluia: for the Lord God omnipotent reigneth. 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. And he saith unto me, Write, Blessed are they which are called unto the marriage supper of the Lamb. And he saith unto me, These are the true sayings of God. And I fell at his feet to worship him. And he said unto me, See thou do it not: I am thy fellowservant, and of thy brethren that have the testimony of Jesus: worship God: for the testimony of Jesus is the spirit of prophecy.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.