A lot of people don't act sexist in real life, but it's easy to accidentally work a potentially harmful or insulting message into a work of fiction. Here are some tropes to watch out for.

Misandrist

A man hitting a woman is scandal, but a woman hitting a man is funny

This is easily the most outrageous trope on the list, especially because it's not confined to the realm of fiction. The more you portray is as okay in fiction, the more it happens in real life, and the more innocent men get slapped in the face and feel like they don't have a right to defend themselves. For your conscience's sake please avoid this fucking trope in your story. If a woman hits a man, it should either be justified because of something evil the man has done, or it should be portrayed as the baseless violence it is (preferably he should hit her back).

Fathers are harsh, mothers are loving

I can already hear you starting to say "But I have statistics here that validate this trope". No, it doesn't matter if it's statistically accurate. Even if trends like this do exist, we have a duty to portray them as false, because that helps erase them - it provides an example of a loving father so that real life fathers don't feel like it's somehow natural or right for them to be the strict one and leave the compassion to their wives. If you think can you alternately solve the problem by just being careful to portray your strict fathers as bad for being strict, that creates a different problem: it's insulting to real life fathers. While this may not be as bad, it is still worth avoiding.

Misogynist

The 100% male cast

The most obvious way to be sexist: simply put very few or no women in the story. Look no farther than some of the Assassin's Creed games for a perfect example of this. In the historical past of the games (which is where most of them takes place), the first game has exactly one notable female character, and she is in exactly one scene. Assassin's Creed: Revelations has one female character also, and she is a rescued love interest. Assassins's Creed 3 has exactly one significant female character, and she dies to motivate her son (see below). In real life, about 50% of people are female.

The damsel in distress

Of course, the trope that women can't hold their own in the game of fiction and need to be rescued by men all the time. I have some tips on how to do this in a non-sexist way here.

The all-male family

This one would be funny it wasn't so sexist. A lot of writers don't seem to realize that you can't have a father and sons without a mother. Sometimes there's at least a handwave (usually that the mother is dead) but even if so it's still in most cases just an excuse to avoid having female characters in the story - remember that everything non-default in your story must be so for a reason, and so there should be a mother unless there's a good reason not to. If you have a good reason to only want one parent around, remember it's an option to have the father be dead.

The female love interest is martyred to motivate the male hero

As with many of these tropes, there is no problem with this in a vacuum, but it happens too frequently without ever being reversed. I (and I believe most men) wouldn't like it if the vast majority of our stories involved the male love interest being martyred to motivated a female hero, so let's not always do it the opposite way.

Mandatory nudity

Men are almost never seen naked in fiction. But female nudity is everywhere, whether it's used to take away the character's dignity (as with princess Leia in Episode 6) or to make her sexually attractive to a presumed male audience (as with queen Ming in Lost Odyssey, or with just about every female character there was in Sword Art Online). Appallingly, we see this trope in real life too: men's formal wear covers their entire bodies modestly, whereas women's tends to be very revealing, particularly at weddings.

'Lady' is used to talk down to women

For some reason 'lady' is almost a derogatory term in some cases. Consider the following scenario (this is pretty much something that happens in Mass Effect 2 and that the player is not given a choice to do anything about): a woman is returning to her house and finds that the police are searching it and there is a guard by the door that won't let her in. She argues with him, but he says, "The law is the law, lady. You can't change it." That sounds natural, doesn't it? But what if the genders are reversed and the guard says, "The law is the law, man. You can't change it." That sounds weird and makes us think, "is there something wrong with being a man?" and rightly so.

Male warrior, female archer/mage/healer or other supportive role

This trope is not sexist in a vacuum, at least if there's only a single occurrence of it, but it's so common that if you have only one pair of characters like this it's probably better to invert the stereotypes. Of course, if there is more than one character of each gender on the hero team, Some Of Each is the way of go. (The reason I consider this trope misogynist and not misandrist is because generally the thinking is that women aren't strong/tough enough to be frontline warriors and not that men aren't intelligent enough to be mages or something. And while there might be some truth to that, it is definitely not as big as it is often made out to be.)

Double-edged

Men are just, women are kind

This is distinct from the misandrist trope that fathers are harsh and mothers are loving, because there the implication is that one is good and the other is bad while here the implication is that both are good, they just specialize in different virtues. And there's nothing wrong with that in a vacuum, it's just that it's become such a cliche that it almost seems wrong when you invert the stereotypes. This is another case where the Some Of Each principle can help you.