A woman should learn in quietness and full submission.
I do not permit a woman to teach or to assume authority over a man; she must be quiet.
Context
Ephesus was the epitome of feminine supremacy in religion! For example, the shrine to Artemis, the great fertility goddess was there. She was worshipped as the source of life. There was confusion about the creation story in Genesis, with Eve being viewed as the liberator of the human race and God as a kind of evil dictator who was trying to keep knowledge from humans. So, the church in Ephesus has some confusion about the role of women in the church and especially harsh rules were needed!
Keywords
hesychia is translated as "quiet" in the passage above. The meaning of the word is probably more akin to "a tranquility and quietness arising from within". It is helpful to see what other passages the word hesychia occurs in:
I urge, then, first of all, that petitions, prayers, intercession and thanksgiving be made for all people— for kings and all those in authority, that we may live peaceful and quiet lives in all godliness and holiness.1 Timothy 2:1-2
We hear that some among you are idle and disruptive. They are not busy; they are busybodies. Such people we command and urge in the Lord Jesus Christ to settle down and earn the food they eat.2 Thessalonians 3:11-12
authentein is translated as "to assume authority over" in the passage. There are many other Greek words translated to mean authority in the Bible, but this is the only time this one is used! So, it is necessary to look outside of the Bible at other literature to find the meaning. It has a wide range of uses, including:
1. to begin something, to be primarily responsible for an action or condition
2. to rule, to dominate
3. to usurp power or rights from another
4. to claim ownership, sovereignty or authorship
Grammar
The grammar of v12 suggests that authentein has been used to explain the KIND of teaching that a woman should not be giving. If that is the case, the alternative translations would be:
"I do not permit a woman to teach or represent herself as originator (or source) of man, but she is to be peaceable."