The number one most quoted verse concerning this is from Ephesians 5.
22 Wives, submit yourselves to your own husbands as you do to the Lord. 23 For the husband is the head of the wife as Christ is the head of the church, his body, of which he is the Savior. 24 Now as the church submits to Christ, so also wives should submit to their husbands in everything.
What exactly this means is very debated among the many Christian groups.
However, these verses are surrounded by others about the hierarchical order of the family and the Church.
21 Submit to one another out of reverence for Christ.
25 Husbands, love your wives, just as Christ loved the church and gave himself up for her 26 to make her holy, cleansing her by the washing with water through the word, 27 and to present her to himself as a radiant church, without stain or wrinkle or any other blemish, but holy and blameless. 28 In this same way, husbands ought to love their wives as their own bodies. He who loves his wife loves himself. 29 After all, no one ever hated their own body, but they feed and care for their body, just as Christ does the church— 30 for we are members of his body. 31 “For this reason a man will leave his father and mother and be united to his wife, and the two will become one flesh.” 32 This is a profound mystery—but I am talking about Christ and the church. 33 However, each one of you also must love his wife as he loves himself, and the wife must respect her husband.
The Scriptures are not particularly clear about what submission is, but it is clear about love in 1 Corinthians 13. What is almost universally agreed is that the husband is "the spiritual head of the household". He is responsible for the spiritual well being of his entire family. Because of that responsibility he is held to a higher standard: to love his wife as Christ loves the Church and readily gave himself up for her. The wife, then, is to respect and trust her husband's decisions, and support him in the very difficult task of being to her like Christ is to the Church.
Chapter 6 carries on with the same principles concerning children and parents, especially fathers.
1 Children, obey your parents in the Lord, for this is right. 2 “Honor your father and mother”—which is the first commandment with a promise— 3 “so that it may go well with you and that you may enjoy long life on the earth.”
4 Fathers, do not exasperate your children; instead, bring them up in the training and instruction of the Lord.
Again, the Father (husband) is given more responsibility and the more difficult task. He is to instruct the children in the ways of the Lord without making them feel demeaned or infuriating them. Anyone who has children knows this is very difficult. Like the wife, children should respect and trust their Fathers' decisions.
Genesis 2 is sometimes quotes as well. After Adam was created, God determined that it was not good for man to be alone so he made Adam a helper: Eve, the woman. The term itself do not necessarily imply a lesser standing, however, it does imply the purpose: the woman is to help the man.
I have heard this said before and I think is sums up the above quite well:
The glory of every husband is that the woman was made for him, but his humility is that he needs her.
The glory of every wife is that the man needs her, but her humility is that she was made for him.