In The Family: A Proclamation to the World, the First Presidency of the Church set out clear guidelines and principles about the roles of husbands and wives, both respectively and together, in the home. Husbands are to protect and provide for the family. Protect in both the physical aspect of keeping them safe from harm, and also in the spiritual aspect. He is the priesthood holder and leader in the home, and is responsible for his family's spiritual well being.
Mothers on the other hand, are to love, teach, and nurture their children. Women have been blessed with a special God-given gift of sensitivity and love. They are responsible for teaching their children goodness and truth. A child grows up to be much the same person as they were taught to be at home when they were young, so it is incredibly important for mothers to teach them things of righteousness and the truth and power of the gospel.
Husbands and wives also have joint responsibilities in the home. They need to work together to support each other in their righteous endeavors in the home. It is close to impossible to create a successful and righteous home if both parents do not work together to create it. That is another reason why it is so important to choose carefully who your spouse will be for eternity. But if both spouses work together than a happy and righteous home to the Lord can be built up.