Sexist language refers to words and phrases that demean, ignore, or stereotype members of either sex or that needlessly call attention to gender. It's a form of biased language. On the surface level, eliminating sexist language from your writing can be just a matter of word choice or making sure your pronouns aren't all "he" and "him."
Look at your pronouns. Have you used "he" and "him" throughout the piece? To revise this out, you can use "he or she," or maybe, if context permits, pluralize your references to use the cleaner "they" and "their" instead of "he or she" and "his or her" in one sentence, as it could become awkward, wordy, and cumbersome. For example, "When a person sells a car, he or she needs to locate his or her title paperwork" could be more smoothly done by revising to plural: "When selling a car, people need to locate their title paperwork."
You can also try and revise out pronouns to be articles. You could locate "the" title paperwork in the example sentence instead of "their" paperwork and not lose any meaning. If you would like practice recognizing and eliminating sexism from writing, see this exercise in eliminating gender-biased language.
Looking for Bias
On a deeper level, you'll want to look at details of the piece you're writing to make sure that it doesn't somehow portray all scientists as men, for example. In "A Canadian Writer's Reference," Diana Hacker wrote, "The following practices, while they may not result from conscious sexism, reflect stereotypical thinking: referring to nurses as women and doctors as men, using different conventions when naming or identifying women and men, or assuming that all of one's readers are men."
Some job titles have already been revised out of sexist usage in our everyday vernacular. You'll probably more often hear the phrase "flight attendant" nowadays rather than the now antiquated-sounding "stewardess" and hear "police officer" rather than "policeman." And people don't use "male nurse" anymore, now that nurses of both genders are a common sight in medical settings.
You'll want to look at the undercurrents in your writing. If you're writing fiction, you'll look at things like, for instance, are the female characters (or male) portrayed as complex people, or are they used just as plot devices, flat as cardboard stand-ups?
Examples and Observations
Ensuring parity is an important topic. Here are some examples of the many sides of the issue, including one where satire helps make the point:
"Questions and criticisms of sexist language have emerged because of a concern that language is a powerful medium through which the world is both reflected and constructed… Some have claimed that the use of generics (such as 'mankind' to refer to both men and women) reinforces a binary that sees the male and masculine as the norm and the female and feminine as the 'not norm'… "
(Allyson Jule, "A Beginner's Guide to Language and Gender." Multilingual Matters, 2008)
Language in Context
The 'language as sexist' prong of language and gender studies has faded in the last two decades… It was soon realized that a word could not unproblematically be derided as sexist since it could in principle be 'reclaimed' by a given speech community (queer probably being the most famous actual example)."
( Lia Litosseliti, Jane Sunderland, eds. "Gender Identity and Discourse Analysis." Johns Benjamin Publishing Company, 2002).
Sexist Language in The Office
Michael: Okay, so what I want to engage us in today is a hardcore discussion about women's problems and issues and situations. Magazines and TV shows and movies portray women as skinny, tall goddesses. Well, look around. Are women like that? No. No, they are not. Points to Pam Even the hot ones aren't really that skinny. So what does that say? That says that you women are up against it. And it is criminal. Society doesn't care. Society sucks. I don't even consider myself a part of society, FYI, because I am so angry over all of this…
Karen: What you're saying is extremely misogynistic.
Michael: Yes! Thank you. That was not necessary, but I appreciate it. And it proves my point: women can do anything.
Karen: I'm saying that you're being sexist.
Michael: No, I'm being misogynistic. That is insane, I'm not being sexist.
Karen: That's… it's the same thing.
(Steve Carell and Rashida Jones, "Women's Appreciation." The Office, 2007)