There are a huge number of organizations who advocate for gay rights. Two are
National Gay and Lesbian Task Force and PFLAG: Parents, Families and Friends of Lesbians and Gays.
There's an extensive list of LGBT (Lesbian, Gay, Bisexual, Transgender) groups around the world here.
Alvin Fritz of University of Washington Libraries has assembled a very comprehensive list of national and international gay and lesbian organizations and publications. His table of contents is as follows (each one is a link):
What groups are you aware of? Were you surprised at the number of organizations listed within these links?