One of the things that becomes clear if you spend enough time around the internet is that there are people who are versed in anti-oppression discourse and people who are not. To the latter category of people, the former can seem like capricious, pretentious, self-righteous, arbitrary judges, eager to exercise their powers to declare you Unfit As A Human Being. To the former, the latter category of people can seem ignorant, rude, and selfish.
(I've found that the best way to find out what category of person you're talking to is to drop the word "privilege" and see how they react. But that's a subject for later.)
Here's the thing: everyone in Category A was once in Category B. Yes, even if they are a queer woman of color--they were probably once kind of douchey to disabled people, before they grew up and realized how society at large fucks up everybody's heads in some way or another. So, this guide is for the Category B people who don't want to be douchebags, but who don't really get what the big deal is about "privilege" and "kyriarchy" and are tired of being accused of being bigots.
You aren't Unfit As Human Beings. You just need to adjust your goggles a bit.
In our society, there is a thing I like to call the kyriarchy. That basically boils down to this: "People like to use any excuse to have power over other people." It's a fancy word for all the different oppressions that go on in the world.
One of those oppressions is sexism. In a sexist society, men are people, and women are not quite as much people. That's the crux of it. I don't deny that men have to pay a price to be considered Proper People. It's not nice. It benefits some men by privileging (there's that word) their needs and desires over those of women, and arranging society so that those needs and desires can be easily filled, but in the end it's not really good for anyone. But men still get to be people in ways that women do not. The patriarchy harms men and oppresses women.
This is the way all oppressions work: by designating one class of people as human, full people, normal, and everyone else as Other.
Men are civilized, rational, intelligent people. Women are overemotional, irrational, flighty beings designed primarily to fill the sexual needs of men.
Then there's racism. White people are civilized, intelligent, benevolent people. Everyone with differently-colored skin or differently-shaped eyes is savage, animal, untrustworthy, lazy, shifty, or backwards, or maybe all of the above and some more.
Of course most people don't openly believe this, at least not anymore. But it's been passed down in society for generations, and it lingers. If you're not in the oppressed group, it's easy not to notice how, but if you are, it's easier to see those little messages everywhere, telling you that you're of lesser value because of something beyond your control. Some people, even of oppressed groups, get by fine, it's true. But just because some women don't feel they have any trouble with sexism doesn't mean it doesn't exist, or the messages aren't there.
People who read up on anti-oppression work learn a lot of terms to describe these situations. One of them is "privilege." If you can go about your life without worrying that the shape of your body, color of your skin, configuration of your mind, or anything like that (the list goes on and on) will cause you regular problems, will regularly make people think about you as different and lesser, you have privilege. Privilege is being able to dismiss an argument about sexism or racism or ableism or homophobia or sizeism or and so on as simply an argument, and not relevant to your day to day life.
Obviously, from a list like that, everyone has some privilege. So "you have privilege" is absolutely not a way of saying that you're a bad person. It also doesn't mean you have an easy life. All it means is that in some ways, and only some ways, society is biased in your favor.
Society is biased in favor of men, white people, able-bodied people, heterosexuals, neurotypical people, cissexuals, upper-middle-class people, and more. When you're in one of those groups, and you start to see the privileges you have over other groups disappearing, it's easy to feel like it's unfair or things are changing too fast, because you're so used to things being in your favor that it seems like any change rearranges things to be biased against you.
An example: I am upper-middle-class. If my family's taxes were increased in order to take the burden off lower classes, it would suck for me, and it might feel like poor people suddenly had an advantage over me. But they don't; I had an advantage over them before, and now it's been addressed a little bit.
Affirmative action (regardless of whether or not you think it's a good idea) doesn't mean that people of color and women have privilege over white men. It means an attempt (whether good or bad) has been made to address existing inequality.
The important thing to note is that this system of privileges and disadvantages is ingrained in society after many generations of building the kyriarchy, and we have only just begun to try to tear it down. Passing situations where disadvantaged groups have momentary advantages do not reverse thousands of years of oppression instantly. Many things that seem like advantages are manifestations of oppression. Men will buy women drinks in bars; this isn't "female privilege," but rather an example of the way women are sexualized (often a man buying a woman a drink thinks of it as a sexual gesture, even if they aren't consciously thinking "I will now buy this woman's sex with a drink"). White people will assume that black people are better at athletics; this isn't "black privilege," but an example of how society teaches that black people are more animal-like and less intelligent.
Here's the most important thing: the more you learn about the language of anti-oppression activism and privilege, the easier it is to talk about without everyone blowing up in a giant ball of hurt and anger.
There are lots of detailed guides on how to have a polite conversation about these topics. I don't have links to any of them at the moment, but if anyone wants to link in comments, I'll edit this to include them. However, that's for later. Here I want to just list really basic stuff that everyone should know and everyone (myself included) forgets sometimes.
1) Have some common courtesy. It is absolutely amazing how much sexist/racist/homophobic/what-have-you behavior comes down to simply not being polite and courteous. I would much rather deal with a conservative Southern gentleman who has been raised with old-fashioned and possibly bigoted ideals, but also understands that you need to be polite to everyone, than with a 4chan addict who has gay friends but still thinks it's hilarious to toss around the word "faggot" as a generic insult.
Society teaches us that there are some classes of people who are less than people, to whom we don't have to be polite. The first key to not being a privilege-fettered ass is shedding this idea. Whatever non-"PC" thoughts might come into your head about a person or situation, you don't have to say them. You just have to remember common courtesy.
2) Be humble. I don't mean be falsely modest and self-effacing. You can take praise! You can admit that you're a good writer or RPer! What you shouldn't do is act like the world revolves around you and your word is law. If someone disagrees with you, whether it's about characterization in a fanfic or the nature of racism, it's not the end of the world. No one is required to adhere to your opinion.
This seems kind of obvious, just like the last point. But it's really important. Just as society teaches us that some people are less than people, it teaches us that some people are more than people, and their statements and opinions are worth more than those of others. Shedding that idea and acknowledging that you are no more and no less a special snowflake than everyone else is crucial.
3) Don't get defensive. When someone points out that you just did something sexist, say that you're sorry, remember it so you won't do it next time, and then move on. If you think they're being unfair, you can go complain a bit to a friend in private. Throwing a fit publicly makes everyone involved look bad and muddies the waters for the next time someone wants to discuss this sort of thing. Apologizing and moving on costs you nothing. And you've learned something, too: doing that particular thing is a bad idea.
4) Remember that other people have different perspectives, and this is okay. A woman is going to see sexism in places that a man doesn't. One woman will see sexism in a place that another woman doesn't. That's okay. It doesn't mean that they're crazy and you need to lecture them about why that sexism doesn't exist. Just nod, politely mention that you disagree if you'd like to, and then move on.
5) Don't place too much stock in your objectivity. Just because you can keep your cool in a discussion of sexism with a woman and she can't doesn't mean you're right and she's a crazy irrational harpy. Just because you can stay calm in a discussion of racism with a person of color doesn't mean you're right and they're scary and angry. It means the subject hits closer to home for them, and it hurts more. Instead of dismissing their concerns because they're overly emotional and too close to the topic, consider that maybe their closeness to the topic comes from experience.
6) Don't do these things to please the PC police or earn cookies from your minority friends. Do it because it'll make your interactions with other people more pleasant for everyone involved. You might even make more friends.