Feminism, broadly put, is a political/philosophical movement that seeks to address institutional differences between men and women in society. When feminism began, of course, these institutional differences were far more significant: women were largely excluded from the workforce, completely excluded from positions of authority, discouraged from voting, unable to get credit cards or personal bank accounts, even obliged to give up their names on marriage (i.e., it was expected that Jane Smith would become Mrs Bob Jones as soon as she said "I do"). Great progress has been made these issues — enough so that the average millennial is shocked hearing about them — but there's still a way to go to achieve the kind of institutional equality that feminism asks for.
Radical feminism extends that line of thought from practical (institutional) concerns to philosophical/metaphysical issues of equality. They don't merely want women to have equal pay, or to break down professional glass ceilings. They want the social, political, and economic worlds to be free of gender biases. Sometimes this is (wrongly) interpreted as a 'gender-neutral' society in which socially constructed concepts of masculinity and femininity don't exist, but the more sophisticated (and difficult) understanding is that radical feminists want a society where women have the same rights, privileges, and abilities to determine when an interaction is or is not gendered/sexual that men have traditionally enjoyed.
Mary Wollstonecraft (way back in the 18th century) said: "I do not wish [women] to have power over men; but over themselves." 'Having power over oneself' means, first and foremost, establishing oneself as a metaphysical equal to others in society, and this is the philosophical conundrum that radical feminism tries to work through.