Where to begin on the subject of feminism.
I guess firstly, to acknowledge that there is actually no definitive description or agreed consensus of what feminism is.
The original suffragettes of the first wave feminism had an easy task in describing their version of feminism. The social and political battles were obvious, concrete and highly persuasive. No need to go over what those extraordinary women achieved - except to remark in passing that I wonder if those women had been shown, say, Miley Cyrus's wrecking ball, as some kind of end result of their movement, whether they would feel disappointed or delighted.

Then you've got second wave, third wave and post-feminisms... there are so many variations and heated debates within each movement, that it is impossible to know where one fits exactly. I have always considered myself a radical feminist, like many of my friends. Yet our opinions on the subject are as diverse as our personalities - which I think is probably extremely healthy. From innumerable interesting and long conversations with feminist friends of both sexes, I have nutted out three main areas of controversy. People tend to fall very clearly on one side of the argument over these  and are likely to defend their positions with rabid vehemence.
(Disclaimer: I've given each side of the argument a generalised type of feminist name - of course it is not accurate and not based on a learned understanding of feminist theory so don't expect a nuanced discussion of the differences between the various schools of feminism here... this is just me categorising my conversations with my friends and finding the closest match I am aware of).

1. The first is gender identity polemics and the range of differences between the sexes:

• There are those who think that biological differences between the sexes are negligible, and that almost all behavioural, emotional or mental distinctions are learned behaviours imposed upon us by the social and political power structures of our society. (see for example Simone de Beauvoir "one is not born a woman but becomes one"). These include gender theorists who usually argue that a two-gender system is neither innate nor universal. In fact they believe that gender (as distinct from biological sex) itself is an entirely constructed imposition that needs to be shrugged off. The state of personally identifying as, or being identified by society as, belonging to either the male or female genders is considered relative to the individual's gender role in society, gender identity, and sexual orientation. For these kinds of feminists an ideal world would be one where bisexuality is the norm, transgender people are extremely common, ambisexuality is a completely acceptable and desirable state of being and where humans are evolving to become hermaphrodites. In other words, a society where notions of gender do not limit or define us as individuals in any way. For the sake of efficacy let's call this group the anti-gender feminists. They are also sometimes known as post-structuralist feminists. (Another good example is Judith Butler's Gender Trouble - click on book covers for more info)
  

• Then there is the middle ground school of thought which believes there are some differences between the sexes, mostly due to the different hormones that race through our bodies, and that these differences aren't necessarily problematic as long as they don't become tools of oppression or inequality from a social, political and economic standpoint. I guess most people would fit into this category. People who are more concerned with ensuring women get equal pay and equal career opportunities and can do everything men are able to do without discrimination. Let's call this school the pro-equality feminists. They often focus on the undesirability of women of being pushed into the housewife role and their heroes are women CEOs who have it all. (see Betty Friedan, for example).

• Then the third group are pro-difference feminists who believe there are important differences between the sexes that are natural and even positive from an evolutionary perspective and should not be overlooked or ironed out. Difference feminists believe differences in gender are basically biological in nature, but are intrinsically (both chemically and genetically) complex and effect processes of thought, emotions and have a huge impact on the whole construct of personality.  (a good example is Camille Paglia). People who think like this tend to fall into two schools- the reverse gender polarity feminists who believe that the sexes are not equal because women are in fact superior to men (and can fall into the trap of female chauvinism very easily), and the gender complimentary feminists who believe that men and women compliment eachother , either as two parts that make up a whole, or as two integral wholistic beings who contain elements of each, and when put together in an ideal way balance eachother (see Margaret Fuller). This third group often contains hippies and metaphysicists.