Two historic views
Complementarian
Complementarians believe that God has distinguished roles for men and roles for women in the church. The two are equal but different. Particularly, this view does not ascribe to women the role of teaching, preaching, or leading, specifically in the church context. This view is primarily supported by a very literalist interpretation of Genesis 2:18, 20b - 22; 3:16; 1 Timothy 2:12; and 1 Corinthians 14:34-35 (see below).
Egalitarian
Egalitarians believe that God has not distinguished roles for men and women in the church. The two are equal and free to serve in whatever function they are called by God. This view appeals to the whole of scripture and makes its case by pointing out the many instances of female leadership in scripture, as well as pointing out some of the textual shortcomings of the complementarian view, to which we will turn below.
It’s worth noting that some people believe in different roles for men and women in marriage relationships but still hold to an egalitarian understanding of a woman’s participation in the church.
Passages that seem to suggest that women should not lead and teach in the church
"A woman should learn in quietness and full submission. I do not permit a woman to teach or to have authority over a man; she must be silent. For Adam was formed first, then Eve. And Adam was not the one deceived; it was the woman who was deceived and became a sinner. But women will be saved through childbearing—if they continue in faith, love and holiness with propriety."
1 Timothy 2:12-15
So much of our interpretation of women in church leadership hinges on the understanding of the word authority in this verse. Here are some critical considerations about the word “Authority”
The Greek word αὐθεντεῖν (authentein) is defined as follows in the Complete Word Study Dictionary: New Testament
αὐθεντεῖν (authentein) - murderer, absolute master... A self–appointed killer with one’s own hand, one acting by his own authority or power. Governing a gen., to use or exercise authority or power over as an autocrat, to domineer (1 Timothy 2:12). See anḗr (435, XI, C), husband.
In 92 occurrences of the English translation "Authority" in the New Testament this word is used only once. While the literal meaning of the word is quite gruesome, the idiomatic use of the word had more to do with barking commands, or dominating, which seems like something Paul would not want to permit (be it a man OR a woman).
The occurrences outside the New Testament are trickier to find because they also are few. αὐθεντεῖν is found in other ancient Greek literature where it is used in reference to violent crimes including murder, suicide, and even child sacrifice. “The Greek orator Antiphon used this word in his legal briefs four times to refer to murder and one time to refer to suicide. Dio Cassius, Thucydides, Herodotus, Euripides, and Philo all used the word in this way” (Braun 1981). In the Septuagint, Wisdom 12:6 uses the word to describe murderous parents.
Others (Catherine Koeger, Michael Green, and John Chrysostom) have found that the word is sometimes used with a sexual connotation in ancient Greek literature, including literature contemporary with the New Testament.
Still others (complementarian scholar Albert Wolters) found that “. . . the word αὐθεντεῖν played a prominent role in Gnosticism (an early Greco roman worldview that often infected the church as heresy); for example it was the name of the supreme deity in the systems of early Gnostics. ”Authentēs is typically translated into English as “supreme power” in works by Early Church Fathers who addressed Christian Gnosticism. This would have close link to the church in Ephesus to whom Paul is writing 1 Timothy since in Ephesus they were greatly influenced by the Gnostic worship of Artemis.
To understand the intent of Paul's words, the context is incredibly important. For example, in both 1 Timothy 2:11 and 2:12 Paul uses the singular term for A woman (one woman), not plural women. Which means there was likely a specific woman who was teaching bad doctrine (likely Gnosticism), and perhaps doing so in an overbearing authoritative manner. Consider the context of the letter as a whole, Paul is writing to Timothy against false teachers in Ephesus (see chapter 1). Also, the very next verse (1 Timothy 2:13) reminds Timothy that Adam was created first. This is not to say that Adam is more important than Eve; that would be completely contrary to scripture. Rather, Paul writes this because in Ephesus at the Temple of Artemis, according to their Gnostic belief the woman came first in creation, then she corrupted the man. This is consistent with Artemis worship in which women dominated men, even with ritualistic castration of male priests, which was still practiced in Paul's day, even though it was made illegal. There are powerful accounts about the early Ephesians church attending to the medical needs of castrated men outside the Temple of Artemis.
Given the historic context of 1st Century Ephesus and the literary context of 1 Timothy, it seems more and more convincing that Paul was addressing a specific problem unique to the Ephesian church, possibly even a specific person within the church. This explains the use of this rarely used, very specific word that we clumsily translated as "Authority". One would be hard pressed to use this one verse as a proof text for what has become traditional historic Christian view of male and female roles and relationship (though that happens all the time).
"Women should remain silent in the churches. They are not allowed to speak, but must be in submission, as the law says. If they want to inquire about something, they should ask their own husbands at home; for it is disgraceful for a woman to speak in the church."
1 Corinthians 14:34-35
It might appear on the surface that a woman is not permitted to teach in the church. But a quick look at the context dispels such an interpretation.
Context: In Chapter 11, verse 5 Paul has already suggested that women can prophesy, so he certainly is not suggesting that women cannot talk, teach, sing, etc. Upon further inspection we see that this is more about questions than about teaching (verse 35 ‘inquire’). The context of this passage is about order in worship. It is likely that there has been disruptive behavior by some of the women in the church.
"But for Adam no suitable helper was found."
Genesis 2:20B
This verse is typically used to suggest that the woman has a specific role, namely ‘man’s helper’. More specifically, this role is defined as a lesser role, not a role of leadership or authority but a role of submission. I.e. the man needed some help so God made a servant for him.
The problem with the suggestion above is that it ignores the broad meaning of the Hebrew word ā·zǎr from which ‘helper’ is derived. It ignores the fact that ā·zǎr rarely denotes a role of submission. In fact, several times it is used to describe God himself (e.g. Psalm 30:10).
"...he will rule over you"
Genesis 3:16
It is imperative to note that this passage represents the results of sin. The man ruling over the woman is not a result of God’s perfectly created order, it is the result of brokenness and is a consequence of sin. The Genesis 3:16 text represents the unwanted result of sin. It is a violation of God’s original plan for unity, equality, fellowship, and community.
To use this passage as a proof-text to prohibit a woman to be equal to men in the church, marriage, etc. is to suggest that the brokenness of the world we live in was God’s design.
Scripture that affirms women in leadership
“Meanwhile a Jew named Apollos, a native of "Alexandria, came to Ephesus. He was a learned man, with a thorough knowledge of the Scriptures. He had been instructed in the way of the Lord, and he spoke with great fervor and taught about Jesus accurately, though he knew only the baptism of John. He began to speak boldly in the synagogue. When Priscilla and Aquila heard him, they invited him to their home and explained to him the way of God more adequately."
Acts 18:24-26
It is clear from this passage that a woman, Priscilla, instructs a man, Apollos. Furthermore, there is nothing in Acts to suggest that this would be unacceptable.
"Greet Mary, who worked very hard for you. Greet Andronicus and Junias, my relatives who have been in prison with me. They are outstanding among the apostles, and they were in Christ before I was."
Romans 16:6-7
This is a very interesting passage. Paul is listing people in Rome whom he wishes to be greeted by the church. In his list he identifies these two people (Andronicus and Junias) whom he calls “outstanding among the apostles”. What is particularly significant here is the person Junias. Junias is a man’s name. However, when looking at all the availible ancient Greek copies of this text, the much more likely choice within the Greek variants is not the male name Junias, but the female name Junia. There is little room for dispute here. Even the Greek New Testament I and every pastor, every scholar, have on our shelves says the female name Junia is the more likely choice. Sadly, translators are often bound to their own biases, which is why it is always best to read those texts in their original language. Which is, admittedly, impossible for most of us. In this case, the translators frequently choose the less likely male name Junias despite the fact that evidence points overwhelmingly to a woman.
The likely choice of the name Junia has some important implications. Not least of which is that it ascribes the gift of Apostleship to a woman. This is yet another example of a woman holding one of the highest roles of authority in the New Testament Church.
"Leaving the next day, we reached Caesarea and stayed at the house of Philip the evangelist, one of the Seven. He had four unmarried daughters who prophesied."
Acts 21:8-9
Here we simply see the designation of Phillips daughters as prophets, thus again showing that there are no roles that are forbidden to women.
Judges 4
In this passage we see the account of the Prophetess Deborah, who is not only distinguished as a leader of Israel, she is one of the few Prophets who was actually a leader of good character.
2 Kings 22
The Prophetess Hulda also stands as another example of God using a woman to prophesy to his people.
Other Women of Note
Leaders of house churches, Phoebe, Lydia, Chloe, and Nympha are named. As owners of homes in the ancient Greek world they would have certainly had authority in their own homes.
Jesus' Ministry
Jesus' ministry was a foreshadowing of God’s Kingdom. He modeled what God’s created purposes were in the world. Where there was sickness he brought healing, where there was death he brought resurrection, where there were demons he drove them out. Likewise, Jesus sought to restore the social order as well. We see this modeled in his respect for women and his love for all those who were looked down upon in society. We see this in his interaction with women as well as in his parables where even women were used as allegories for God himself (e.g. the parable of the lost Coin). Christ brought redemption to all people, all creation, and all social structures, including patriarchy. Some may describe Jesus as progressive, but he was not a feminist. Jesus was for all people and he was seeking to bring restoration to the created order. That advancement of his kingdom is meant to restore all things back to the way he created it to be.