The comprehensive command to do all things in the name of the Lord (3:17) is now applied to everyday household relationships. The Lord wants our social relationships to be sanctified. Paul first deals with marriage and family.[1]
Marriage (18-19): Christ issues a duty for the husband and a different duty for the wife. She is to willingly place herself under her husband's authority.[2] And she is to do this because it is fitting in the Lord. Christ is the reason for her submission.[3] As for the husband, he has no right to be a tyrant. Instead, he is to love his wife. Also, he must not harbor any bitterness. Those closest to you can upset you the most,[4] but the Christian man must forbear and forgive.[5]
Family (20-21): The children are directly addressed.[6] They are to obey their parents in all things, because "this is well-pleasing to the Lord." Total ("all") obedience is how children can please the Lord. Christ Himself set an example of this when He was a child (Luke 2:51). The father, on the other hand, must not abuse his authority. He can wrongly cause his children to "lose heart." His God-given role is to train them (Eph. 6:4), not to ruin them. Discipline should be fair and firm (Heb. 12:10-11), and it should come from love (Prov. 13:24).
[1] These, plus slavery, were common in the first century household. Same groups appear in Eph. 5:22-6:9.
[2] The imperative "be subject" is also "submit to" (ESV, NKJV, NET). Women are to actively pursue submission, or literally, "place themselves under." This is a willing placement of herself under her husband's authority.
[3] "As is fitting in the Lord" explains why she submits. This parallels the explanation in 3:20 ("for this is well-pleasing to the Lord") and lines up with the comprehensive scope of Eph. 5:24. The wife submits because it is Christian (not cultural) to do so. This said, God's authority always trumps human authority (cf. Acts 5:29).
[4] This is why in the church of God there is need for patience, forgiveness, love, etc. (Eph. 4:31-32; 1 Pet. 4:8-9).
[5] Cf. 3:13. Instead of bitterness, God wants the husband to live with her with understanding (1 Pet. 3:7).
[6] In this household context, the "children" are young and in need of raising up (cf. Eph. 6:1-4). The duty of married adults is their marriage (3:18-19; 1 Cor. 7:3; 11:3) and the care of their parents (1 Tim. 5:4, 8).