This is an almost infinitely complex question, and one that I don't think really has a satisfying answer. However, I can provide some context to your question that I think will be helpful. First off, let's ask the question 'what do we mean by "gay?"' 'Who is gay and who is not gay, in a historical sense?' "Gay" as an identity category, doesn't really exist before the 1940's-ish. Even "homosexual" doesn't really exist before the late 1800's. Just like everything else in history, notions of sex, sexuality, and identity have been constantly changing over time. Some interesting things started to happen around the turn of the last century that put the groundwork for today's notions of homosexuality into motion. People started to think about their sexuality as a part of their identity, as opposed to simply something that a person did. In the past (in a very broad, general sense) what role you played in a (male) same-sex encounter was the discerning factor, not whether or not you had them. The "active" or penetrative partner was considered to be the "man's" role; men who took this role, regardless of the gender of their partner, were "normal" "masculine" men. The "passive" or receptive partner could take different forms. He might be a younger man/adolescent boy (who might well grow up to take the active role in his adult sexual encounters) or he could be a "fairy" or "pansy" - an adult male who adopted a feminized persona (a "flaming queen," if you will.) Fairies/pansies were considered effectively the same as women (particularly female prostitutes. In early 20th century America, it was considered quite normal to seek out the "services" of a fairy, especially in the case of fellatio, which female prostitutes were often unwilling to perform.
As I said earlier, this dichotomy starts to shift around the turn of the century (although it was gradual; different, even conflicting notions often coexisted.) A group of men (largely middle-class) started to identify as "homosexual" or "queer" (VERY different from the 1990's definition of queer (which is itself different from today's definition of the word, but that's a discussion for another day.)) These men centered their identity around their attraction to other men, not around their role in sex, or around their gender identity (fairies often rationalized their attraction to men based on their feminine gender identity.) This new identity category helped create another identity category - the heterosexual. Over the course of the first few decades of the 20th century this dichotomy of "gay" and "straight" became the norm ("gay" takes over as the preferred term roughly around the 1940's.)
With that context in place, I'm going to attempt to actually answer your question.
Understanding the change in sexual identity categories is very important. In order for there to be a homosexual rights movement, homosexuals had to exist, and exist in a way that was considered abnormal. I would say that this whole period of change, from the mid-late 19th century to the 1940's, was a time in which the homosexual rights movement was beginning. That being said, I think it's fair to say that the homosexual rights movement - in a form that is recognizable to us now - did truly start in the mid 20th century.
I hope that helps; I'm happy to answer any specific follow-up questions!

EDIT: I realized I forgot the important caveat that my knowledge is primarily US-based. Also, I'll cite George Chauncey's Gay New York as my main source.