A man is the head of the household. To put it another way, the man is the patriarch of his home. He has responsibility for his home, as well as authority over his home.
Today, there are some in "complementarian" circles who seek to limit a man’s headship to his wife, and then claim the husband and wife are co-heads of their children. But this simply does not work biblically. Yes, the man is clearly the head of his wife (cf. Eph. 5:21ff; 1 Cor. 11:3), and this headship traces back to the pre-fall situation (e.g., the whole human race derives its name from the man, Adam names his wife before the fall, and after the fall the Lord calls out to the man to give an account).
But there are many indications that the man should be regarded as the sole head of his household as a whole. (I am fond of saying “Anything with two heads is a monster.”) There is more than I can convey here, but I will suggest a few lines of argument. Paul commands fathers to raise up children in the fear and nurture of the Lord; obviously wives/mothers are also involved, but the man is responsible for the task (Eph. 6:4; cf. Genesis 18, where the task of directing the household in the ways of righteousness is given to Abraham, not Sara). A man can speak for and act on behalf of his whole household (cf. Josh. 24); a woman never speaks for her whole household in this way. The Bible repeatedly uses the formula “the house of….” and it is always the man who is named. We read about “the house of Abraham” but never “the house of Sara,” or even “the house of Abraham and Sara.” In Exodus 6:14, the formula is “the heads of their fathers’ houses.” Psalm 128 is written entirely from the man’s perspective. He is at the head of the table with his family surrounding him; according to the psalm, it is his table, his food, his wife, his children, his grandchildren. In 1 Timothy 3:4-5, a man is judged in his fitness for church office in part by how well he rules his household. This makes no sense if his wife is co-head. It only makes sense if the man has both authority over and responsibility for his household as sole head. In Numbers 30, a man can nullify the vows of his wife or daughter. And so on.
None of this negates the wife’s real authority within the home. For example, the 5th commandment requires children to honor both parents. In the book of Proverbs, the young men is instructed to listen to his mother’s wisdom as well as his father’s. But this does not make her co-head. A vice president has real authority, but that does not make him co-head with the president. Likewise, the wife has real authority, but how she uses her authority is to be in submission to her husband.