Homosexual
A "Homosexual" is a person, of either (any) gender, who is emotionally and sexually attracted exclusively to people of the same gender.
There are multiple words for the trait. "Gay" is gender-neutral, but more likely to apply to men. "Lesbian" applies exclusively to women; this is a reference to the Greek isle of Lesbos, where poetess Sappho kept a collection of women with whom she was enamored ("Sapphic relationship" comes from this as well). "Queer" is gender-neutral, and has often been used as a derogatory slur, but is now being reclaimed by the radical queer movement. Then there's "homosexual" itself, but this word can carry negative connotations (not to mention five syllables) and is avoided outside of technical speak; the shortened version, "homo," is mostly used as a slur, as are "faggot" and "dyke."
Current statistics claim that about one person in ten is homosexual. This has contributed to the historical view that it is unusual at best, a serious deviation at worst. Until 1973, the Diagnostic and Statistical Manual of Mental Disorders, basically the official textbook for abnormal psychology, listed homosexuality as a mental disorder [1]. Just about the only term that has escaped the pejorative label is "Bi", as in "Bisexual," meaning "attracted to members of both sexes". This is partially because the term is relatively new, as is the idea that sexual orientation is a spectrum; and it's also because bi people are still willing to sleep with people of the opposite sex, which is a big point in its favor.
Since homosexuality is harmless as far as sexual abnormalities go, why is there so much hostility towards and/or disapproval of it? One interpretation has to do with conditions people have lived in through history. In the past, child mortality rates were horrific; an average of one child in two died before the age of five[2]. Death by Childbirth was also a major risk, which was even worse because it killed not only the mother, but every child she might have had thereafter. Long story short, you wanted every able-bodied male and female available to be involved in the process of continuing the species. Men and women not interested in reproducing (because they wanted to get busy with their own sex exclusively) added nothing to the process. It should be pointed out that in most cultures, however, having gay sex on the side was okay (in fact, sometimes man-on-man sex was considered a virtue), or given a blind eye, as long as you were still pumping out babies.
Many animal species include homosexual members, just like humans do; and since it's such a widespread phenomenon, biologists theorize that homosexuality in a population must be an adaptive trait despite the stigma. When the population equals or exceeds the food supply, homosexual (and asexual) members of a population can contribute to the well-being of the community, but do not add more children. The end result is that the children of the straight and bi members of the population have a better environment and a better chance of surviving because there are more people to support each child. Infant mortality goes down, the next generation is smaller but healthier, and the genes for homosexuality are passed down to the next generation through their siblings, who are more likely to survive because of the improved environment. If this theory is correct, improved acceptance of gays and lesbians should eventually help mitigate the problem of overpopulation in crowded countries.
The Squick factor is important too. Homosexuality is a concept many straights do not understand, and it's always been human nature to fear/hate the unknown. Men who engage in same-sex relations, especially those on the "receiving" end of anal intercourse, are considered unmanly in most modern cultures, and sick weirdos for not appreciating the fairer sex. Suggesting that a man is gay is often the worst insult against a man in any language. For lesbians, it's a little different but not by much. Girl-On-Girl Is Hot, and it's okay to ogle women making out, but seriously acknowledging a relationship between two women (ie, women who don't need love, or perhaps sexual fulfillment, from a man) is a no-go. And this is where you get cases of people trying to beat the gay out of men and rape the gay out of women.
Historical Context
The term "homosexual," and indeed the entire idea of sexual orientation, is much Newer Than They Think. The first time "homosexual" appeared in print was 1869. Now, this is not to say that there have not been same-sex lovers since basically the beginning of time; in fact, as homosexual behavior is found in animals, it actually predates the human race. It's simply that, as explained earlier, humans placed a huge emphasis on making babies. As long as you were doing that, nobody cared what, or who, you did in your spare time, but it wasn't a lifestyle you could adopt, or identify with. You weren't gay; you weren't straight either. There was no category. As such, going to just about any historical figure and trying to describe them as gay or straight is inappropriate, since such social constructs didn't exist yet. Whether or not William Shakespeare liked boys, it's an absolute guarantee that he didn't think of himself as being gay. Like any good Englishman, he sired issue on his wife, regardless of what he would have liked to do if given the chance. Of course there were people who saw themselves as homosexual in the modern sense of the word, but they often received the same kind of ostracism that goes on today--even though most of them would have had to be rich, powerful, respected or priveleged just to get away with it. They were considered, well, deviants.
Gender Politics
Your sexual orientation describes whether you're attracted to the same gender as yourself, the opposite gender, both, or neither. It's important to note that gender (whether you personally identify as a man, woman, or neither) is not the same as your sex (whether your body is male or female). It's possible, for instance, to be an exclusively straight male but also swing over to the feminine side of the spectrum; Eddie Izzard, who claims to be this sort of person, has described himself as a "male lesbian". For the ladies, you have characters like Shannon Beiste from Glee, a rough and burly female football coach who has no sexual interest in other women. Neither have any desire to change their sex, it's just their personality. Furthermore, transsexuality has nothing to do with homosexuality. A female-to-male transsexual who dates women is not a lesbian, for example, but a straight man, and is just as likely to be gay (attracted to other men) or bi.
The problem is that, in Western culture (and media) especially, gay people are often stereotyped as wanting to be the opposite sex, since many act in ways contrary to their gender. In other words, sexual orientation and assigned gender are treated as though they are not separate issues, despite half a second's thought indicating that of course they are. This is why the Butch Lesbian and Camp Gay stereotypes are so prevalent, though (for men) the Manly Gay stereotype is slowly starting to make headway, especially as it (partially) redeems gayness in public eyes by adhering to masculinity, a quality that is much prized in most cultures. This is also why transsexuality gets bundled up in this issue, despite being loosely relate at best: the media like to stereotype it as being some sort of "ultra-gay" condition.
Modern Times
Today, there are major stereotypes going on, some of them documented on this wiki, about the kind of person you are if you're a homosexual. We've talked about Manly Gay and Lipstick Lesbian, as well as the obvious Camp Gay and Butch Lesbian. Invisible to Gaydar is somewhere in the middle. Being bi just means being awesome; it's become rather trendy today to identify yourself as bi, regardless of whether or not you would enter a serious relationship with the same gender. Having said that, as the Bi the Way trope remarks, a lot of times the media don't know how to deal with bisexuals. And let's not even get into Asexuality or anything more complicated than that. But the point is that, whether rightly or wrongly, homosexuality is perceived as being about more than just who you sleep with; it's thought of as being a lifestyle.
Today, being homosexual is also about politics. As with Gun Control, homosexuality, its legality, its normality and its social acceptability is a Single-Issue Wonk for a lot of people on both sides of the debate. In modern American politics, for example, it's valid to ask, "Would you vote for an African-American presidential candidate just because he's African-American, regardless of his actual platform," because some people would actually answer No (or Yes) to that question. A politician's stand on homosexuality can be a similar deal-breaker.
Even better, there are scriptures in various religious texts condemning homosexual acts. This of course raises its own questions: Why are those condemnations there? Is it to encourage reproduction, or does <Deity Of Your Choice> actually consider it evil? Does the passage of scripture actually mean what you say it does, or is it being taken out of context? Is it just because, before condoms and penicillin, STDs were much more of a problem and monogamy was the best way to reduce their spread? Some people refuse to ask these questions on principle. Indeed, they tend to focus on anti-gay scriptures while ignoring those that condemn their own vices, simply using religion as an excuse for pre-existing hatred.
The point is that this is an issue where personal morality, religion and politics all intersect, and if you talk about it openly someone might ask you for a political justification to what you had thought of as a purely-religious opinion (or vice versa). To call it a mess would be an understatement.
The growing acceptance of homosexuality as a lifestyle does bring up one issue that, for good or ill, even open-minded straight people could consider problematic: the idea of single-gender parenting. What are the implications on childhood if a child Has Two Mommies or Has Two Daddies? Religious conservatives would have you believe that this will result in all manner of perversions, like child abuse, Mind Rape and bestiality. What little scientific research has been done, however, suggests that if anything the opposite is true: homosexual parents tend to be more settled, more sane and more involved as parents--in general, better. Supporters like to point out that gay couples don't get accidentally pregnant quite as often as straight ones; they have to take very active steps and thus have plenty of time to prepare (or back out). They are also a great boon to the adoption industry. Thus far, claims that open homosexuality will bring about The End of the World as We Know It are either being subverted or inverted.
Politics aside, the important thing is that homosexuality has existed for eons and will not be going anywhere any time soon. Gays are as diverse in interests and mannerism as any other group, and even though they continue to face persecution in many parts of the world, more and more countries are coming to the conclusion that gay rights are human rights not bound by any religious or moral doctrine.