In the not-too-distant past English used the pronoun “him” to refer to individuals of unspecified gender in phrases such as “Everyone is entitled to his own opinion.” This understandably annoys some people, who prefer “Everyone is entitled to his or her own opinion” or some other alternative. Sometimes the problem can be avoided by making the antecedent plural, as in “All people are entitled to their own various individual opinions.” Another option is to adopt a sort of jump-ball rotation, alternating between “he” and “she.” Of course, this requires a bit of bookkeeping for balance, and it can lead to such difficulties as “From each according to his ability, to each according to her need” or vice versa. There doesn't seem to be a single solution that makes everyone happy, me included.
A lot of people use a singular “they,” as in “Everyone is entitle to their opinion.” This draws the obvious objection that it’s improper to have a plural pronoun refer to a singular antecedent.
The best defense I’ve yet read of the singular “they” is on the Oxford English Dictionary blog here. It points out that the singular "they" has been used for centuries, and moreover, there's a good case to be made that a singular they is no more objectionable than the singular "you." Quoting the blog post:
In 1660, George Fox, the founder of Quakerism, wrote a whole book labeling anyone who used singular "you" an idiot or a fool. Eighteenth-century grammarians like Robert Lowth and Lindley Murray regularly tested students on "thou" as singular, "you" as plural, despite the fact that students used singular you when their teachers weren’t looking, and teachers used singular you when their students weren’t looking.
Note, by the way, that even in modern English, plural verbs are used with you (“you are” and “you have” as opposed to “you art” or “you hast”) even when the antecedent is singular. That pretty clearly hints that “you” is a plural pronoun. Note that people similarly say “they are” and “they have” when using the singular “they.”
The blog post also mentions that “The New Oxford Dictionary of English (1998) not only accepts singular they, they also use the form in their definitions.” The post concludes, rather emphatically,
Former Chief Editor of the OED Robert Burchfield, in The New Fowler’s Dictionary of Modern English Usage (1996), dismisses objections to singular they as unsupported by the historical record. Burchfield observes that the construction is ‘passing unnoticed’ by speakers of standard English as well as by copy editors, and he concludes that this trend is ‘irreversible’. People who want to be inclusive, or respectful of other people’s preferences, use singular they. And people who don’t want to be inclusive, or who don’t respect other people’s pronoun choices, use singular they as well. Even people who object to singular they as a grammatical error use it themselves when they’re not looking, a sure sign that anyone who objects to singular they is, if not a fool or an idiot, at least hopelessly out of date.