Although the question could go all over the place I am looking to know how evangelicals define the role of a man and of a woman from the Bible.
What are the verses they use? Is there a short description of the different roles?
Note: This is typically a fairly volatile issue. My answer attempts to identify what the Bible has to say on this subject.
It is certainly important to understand the reason for which God made us, including the reason for which God made us as beings with gender. We can only experience the fullest joy when we understand who we are and our unique purpose. God's desire for both men and women is for them to experience the highest joy.
It would certainly seem obvious that there are fairly significant differences between men and women. It is also important to note that differing roles does not indicate differing value or intelligence of honor. Yet, God did create us as beings with gender, so it behooves us to seek to understand why He chose to do that and what that might mean for us.
Gender Roles in Creation
The Bible does not explicitly define gender roles, but it does identify points that are instructive. The creation of man and woman seems to shed a good bit of light on things, including the following.
Gender Roles in the Fall
The Fall of mankind also provides some interesting information that either resulted from the Fall or was already inherent prior to the Fall, but was just identified after the Fall.
Gender Roles in Marriage
Paul identifies the idea of "headship" for the husband, among other things.
Neither male or female
Much is sometimes made about Galatians 3:28 where it indicates there is neither Jew nor Greek, slave or free, male or female. Some would suggest that all gender roles are, in Christ, rendered obsolete because of this. However, it was Paul who wrote that, and he is the one who comes under fire the most for making gender distinctions in his writing. Galatians 3:28 is referring to our standing in salvation. Jewish believers have no better place before God than Gentile believers, just as men have no greater standing in salvation than women.
So, the Bible does tend to reveal a distinction in roles. Men were created first and given work to do. The woman was given to the man in marriage as well. Women seem to have been specifically created for relationship and, in marriage, are specifically instructed to be submissive to the headship of the man, who has the responsibility to treat her with sacrificial love.
The biblical (at least, imo) view of gender roles is, in short, that the man symbolizes Christ and the woman symbolizes the church.
Delving in a little bit deeper, Eve was created as a "helper" for man, a "suitable partner" - not as a creature inferior, though, which I think could be inferred from the fact that Eve was made from the rib of Adam.
But for Adam no suitable helper was found. 21 So the LORD God caused the man to fall into a deep sleep; and while he was sleeping, he took one of the man’s ribs and then closed up the place with flesh. 22 Then the LORD God made a woman from the rib he had taken out of the man, and he brought her to the man.
In Ephesians, Paul says:
Wives, submit yourselves to your own husbands as you do to the Lord. 23 For the husband is the head of the wife as Christ is the head of the church, his body, of which he is the Savior. 24 Now as the church submits to Christ, so also wives should submit to their husbands in everything. 25 Husbands, love your wives, just as Christ loved the church and gave himself up for her 26 to make her holy, cleansing her by the washing with water through the word, 27 and to present her to himself as a radiant church, without stain or wrinkle or any other blemish, but holy and blameless. 28 In this same way, husbands ought to love their wives as their own bodies. He who loves his wife loves himself. 29 After all, no one ever hated their own body, but they feed and care for their body, just as Christ does the church— 30 for we are members of his body. 31 “For this reason a man will leave his father and mother and be united to his wife, and the two will become one flesh.”[c] 32 This is a profound mystery—but I am talking about Christ and the church. 33 However, each one of you also must love his wife as he loves himself, and the wife must respect her husband.
I think the key phrases here are "love his wife" and "respect her husband" - the woman has a supporting role to the loving, caring, providing husband; both epitomizing Christ and the Church. (See Hosea for great imagery on this)
I know this is in the context of marriage, but generally I think it still applies - man providing, woman helping/submitting, especially since this is the general role given at Creation.
Also, in the context of the church, there's some information provided by Paul in 1 Cor. 14:34
"women should remain silent in the churches. They are not allowed to speak, but must be in submission, as the Law says."
Ultimately though, these are minor distinctions: in Christ there is neither male nor female; we are all equally sinful but equally made righteous by the blood of Christ.
I will answer this question simply by presenting the actual roles of women in Scripture:
Based on this list, it is clear that women (and men) are called to many different types of roles. However, when people talk about "gender roles" in the church, the discussion largely looks similar to the posts above. This is unfortunate, because as evangelicals we believe it is important to take all of Scripture into consideration, but people tend not to think of actual women's roles as relevant to "gender roles." And while Scripture may demonstrate lots of gender roles, based in the culture, that we would consider abusive in our culture, the fact that God called and used women beyond those patriarchal roles indicates to me that those roles are made by humans and not the intention of God.
A minor note on Eve being a "helper" as mentioned in one of the responses - The female (not yet called Eve) was created as a helper to the male the way God is described as a helper to humans. The Hebrew word used for Eve is most often used as a word to describe God helping humans - almost 20 times in the OT. The Hebrew word is more like "rescuer" and implies strength. Genesis 2 seems to demonstrate that the male was lonely, and God sent a rescuer, the female. This English translated word is used as a support for the idea of hierarchy, but the people I know who hold this understanding of "helper" do not believe that God is under males or that males have authority over God. They don't believe God is the sous chef. Yet to believe this is what the word means for females implies the same meaning for God.
As you can see, as evangelicals, we do not hold anything close to a monolithic view, there is a great deal of diversity on many non-central issues!