Christianity Today has an interesting post here entitled "The Complementarian Women Behind the Trinity Tussle". Ultimately, at issue is the question of Christ’s relationship to God the Father. Some argue that the Son is eternally subordinate to the Father, while others say the Son was subordinate in his earthly life only.
(For me, there are several verses that indicate the eternal subordination of the Son to the Father - i.e., 1 Cor 15:24; 1 Cor 11:3; etc.)
Subordination tends to make one think of a lower rank or a subservient position. But to properly understand subordination in the Trinity, it is important to grasp the different types of subordination. The Biblical or orthodox view of the triune nature of God acknowledges an economic subordination in the Trinity but denies the heretical view of an ontological subordination.
What does this mean? Simply that all three Persons of the one Godhead are equal in nature. God the Father, Jesus the Son, and the Holy Spirit all have the same divine nature and divine attributes. Contrary to the teaching of many cults, there is no ontological subordination (I.e., no difference in the nature of the three Persons of the Godhead). This means that Jesus is not a lesser god than God the Father, etc.
Scripture indicates an economic (or relational) subordination within the Trinity. That is to day - the three Persons of the triune Godhead voluntarily submit to each other respecting the roles They perform in creation and salvation. So, the Father sent the Son into the world (1 John 4:10). These roles are never reversed in Scripture: the Son never sends the Father. Likewise, the Holy Spirit is sent by Jesus and “proceeds from the Father” to testify of Christ (John 14:26; 15:26). And Jesus perfectly submitted His will to the Father’s (Luke 22:42; Hebrews 10:7).
Economic or relational subordination is simply a term to describe the relationship that exists among God the Father, God the Son, and God the Holy Spirit. Essentially, economic subordination within the Trinity refers to what God does while ontological subordination refers to who God is.
All three Persons of the Trinity have the same essence and nature, but each One has different roles or activities when it comes to how God relates to the world. For example, our salvation is based on the Father’s power and love (John 3:16; 10:29), the Son’s death and resurrection (1 John 2:2; Ephesians 2:6), and the Spirit’s regeneration and seal (Ephesians 4:30; Titus 3:5). The different tasks that we see the Father, Son, and Spirit perform are the result of the eternal relationship that exists among the Persons of the Trinity.
Correctly understanding subordination in the Trinity helps us avoid several false teachings about the nature of God. In the end, having a right view of God is important so that we do not manufacture a god in our own image.