The slang term “fag hag” is used to describe a woman, usually straight, who has a close friendship with one or more gay men. This slang term is a bit slippery and rather curious, since it combines two epithets which are usually considered offensive, and it was in fact originally used as an insult. Today, the term is usually used self-referentially, or by members of the gay and lesbian community, and use of this slang by outsiders may be viewed as only marginally appropriate.
Fag hags may hang out with homosexual men for any number of reasons. Some women simply happen to befriend gay men because they feel that they have a lot in common, while others may actively cultivate friendships with gay men because they view these relationships as safe from sexual drama and tension. Some people accuse fag hags of being secretly in love with their gay buddies, suggesting that women befriend gay men with an eye to converting them.
The slang term “fag” for a gay man is generally deemed highly inappropriate. It is derived from “faggot,” another slang term for a gay man, and it has been in use since the early 20th century. “Hag” is an offensive term used to describe women, usually in the sense of women who are ugly, cruel, old, or aggressive. The combination of both of these offensive pejoratives in a single epithet which is often used affectionately is a bit confusing. "Fag hag" probably reflects the desire of the gay and lesbian community to co-opt terms which have historically been used offensively.
You may also hear a fag hag referred to as a fruit loop or homo honey, illustrating some other examples of co-option of offensive slang terms. People who enjoy hanging out with gays and lesbians in general may be called fruit flies, while men who establish friendships with gay women are known as Dutch boys, for reasons which remain obscure. A number of stereotypes surround fag hags, belying the diversity of friendships between gays and lesbians and the straight community.
As in the case of other terms which have very negative connotations, it is important to tread carefully when using “fag hag” to describe someone. As a general rule, unless a woman explicitly identifies herself as a fag hag, the term should not be used. People who are not allies or members of the gay and lesbian community may want to avoid this particular piece of slang altogether.