A good example is the debate about equal pay. Feminists fight for equal pay for equal work. Their opponents say that we already have equal pay for equal work, but that women choose to do less paid work, or choose lower-paying careers, or don't ask for promotions. Both sides agree that we need equal pay for equal work, but they have a different belief about whether women's rights need an advocate right now.
A quick Google search gives a nice definition of feminism:
The advocacy of women's rights on the grounds of political, social, and economic equality to men.
A feminist advocates for women, specifically, because they see problems with the way women are treated. They call these problems "patriarchy," and there's an enormous amount of scholarship on how sexism and patriarchy affect everyday life. As a psychology student, I saw a lot of evidence of negative stereotypes towards women, so I became a feminist. I believe that God speaks to us every day through men, women, and children, but that almost all of us listen better when a man is speaking. Hence, men control most of the world's governments, churches, and businesses. Even in family life, many women do not have an equal voice in family decisions as the scriptures direct for all Priesthood councils.
So, I'm a feminist.
"But," someone says, "Feminists are so angry at men! Men have issues, too. Who's looking out for men who are abused, or mocked, or hurt?"
This is why egalitarianism is important. Obviously, everyone deserves to be treated like a human being. An egalitarian is open to discussing the needs of anyone and everyone who needs help, regardless of whether they're more disadvantaged than the person standing next to them. If I understand this verse right, the scriptures openly endorse egalitarianism:
[God] denieth none that come unto him, black and white, bond and free, male and female; and he remembereth the heathen; and all are alike unto God, both Jew and Gentile.
Since I believe that women's issues are in need of special advocacy, and I am also open to discussing men's issues, I am an egalitarian feminist.
I propose the following three principles for egalitarian feminism:
1. Standing up for women's rights in every sphere of life.
2. Openness to discussing men's issues, as well as all other social issues.
3. Fighting against blanket statements and stereotypes about either gender.
I propose that egalitarian feminism can advocate for women without being angry at men in general, acknowledging that patriarchy is sustained by both men and women. Egalitarian feminism blames individuals for their crimes, not the victim or the criminal's demographic group. It calls on everyone to consider whether they are treating women respectfully, and it is bold in pointing out real problems in the education and treatment of both men and women. Egalitarian feminism holds that men are human beings, capable of making their own choices, and argues that feminism is liberating to both genders.
Men, we need to stop using "girly," "sissy," or "gay" as an insult. We need to admire sensitivity and empathy in everyone, as God does. We need to respond to statistics about male violence with solutions, not defensiveness.
Women, we need you in politics. We need you in academia. We need you in our personal lives.
Egalitarian feminism does not argue that men and women are the same (that's equality feminism). It does not argue for abolishing gender entirely. Instead, it argues that men and women, together, can choose to treat women better than they are being treated right now.
As a final note, my lovely Latter-day Saint audience, please note that "patriarchy" and our scriptures' "patriarchal order" are not the same thing. The patriarchal order states that women and men lead their families together, as equals, with somewhat different roles; in patriarchy, the man is in charge, with freedom to command and abuse however he likes. Patriarchy and the patriarchal order are exact opposites, despite their similar names. For more on this, as well as what it means to "preside," go study D&C sections 107 and 121. Even the Prophet is required to make decisions in unity with his counselors.