1. Feminists are just women who want female supremacy.
This is wrong in two ways. First, it assumes that feminists are all female and, second, it assumes that it's about giving more power to either one of the two sexes. Feminism is about equality of the sexes, that's pretty much the very core of it and to suggest it's about favouring women is plain wrong. In fact, feminists do indeed try and help males as well in the areas where they are at a disadvantage. For example, in divorces cases, the child is more often than not given into the custody of the woman than the man. 70% of the time, the woman is given custody of any children after a divorce, simply because the law favours her as a 'mother' and therefore as a better parent. Feminists are therefore against this as, clearly, men and women are not treated equally here.
2. In the past when women were treated unfairly, feminism was needed, but now they're equal and it's just pointless.
Anyone who believes this to be true has either a naive or oblivious outlook on things. No matter how you look at the facts, men are still treated better than women. There are small examples of this, and big examples. For a small example, I bet you can't spend a day browsing the internet without coming across a silly 'make me a sandwich' or 'stay in the kitchen' joke like this (these jokes don’t have a male equivalent). For a big example; even today a woman who does a job will probably be paid less than a man who has the same job... Which is absolutely shocking, really.
3. Women say 'sexist' things about other women too, so how can it be sexism?
This is, perhaps, one of the silliest arguments. Now, let's take the argument of 'sluts' for this one. It is sexist and anti-feminist to call somebody a slut because it is a word which is inherently connected to a double standard against women, namely the old "If a woman dresses suggestively and has lots of sex, she is morally corrupt, whereas if a man does it, there's something wrong with him" when, really, so long as the sex is conducted nice and safely, what on Earth is wrong with two consenting adults having sex together? Absolutely nothing. Yet a woman who has had lots of sexual partners will be criticised for being a 'slut'. The argument against this is that other women are probably just as likely to call a woman a 'slut' as men might be and so therefore it isn't sexist. But that's wrong; women can be sexist towards other women just as men can! Just because you are a member of a group, it doesn't mean that nothing you say can be offensive towards the other members.
4. Feminists are just sensitive. Everybody gets insulted about things; they're just making a fuss about it.
Well, as I mentioned before, feminism isn't just about insults, jokes and the language, women are, on average, paid less than men for the same jobs (and here's another source for that). And there are lots of other examples too, for example, street harassment. The Twitter page EverydaySexism is completely full of Tweets from women who have had creepy, sexual, objectifying things shouted at them on the streets or who have, in worse cases, been physically touched or grabbed by random men. I would absolutely hate it if I came across things like that directed at me on a regular basis, and it's another clear example of how women are not equal in our society. From the point of view of the men who do this, it's perfectly fine, because that's just the way they view women as a result of the sexism in our society. This is something I've observed in real life too, several female friends of mine have given me anecdotal stories about it, so it is clearly something that affects a very large number of women.
I also see this brought up quite often too, only to be followed with a point along the lines of "Well, a lot of women are asking for it!" which, again, is a very sexist comment and, in fact, is quite damaging to both sexes. It suggests that, when a woman is wearing a revealing outfit, a man is suddenly relieved of all responsibilities and transformed into a lust powered beast that can't control or be held accountable for whatever comes out of his mouth (which, no man ever is). It's basically the same as if a school bully said "Well, it's not my fault I beat up Dwain, he was answers all the questions in class, he was just ASKING to be hurt!" when somebody does something to somebody, they are responsible for it; it is never the other way around.
So, I hope that clears up any misconceptions people may have about feminism, and that it was also, at least, slightly interesting!