fighting for these days. The answer in the US is the right to not get fired, evicted, or otherwise discriminated against by employers, landlords, and others based on sexual orientation. Similar to previous movements to eradicate such discrimination based on race, ethnicity, religion, gender, etc.
Given the blatant anti-gay laws and practices in many parts of the world, there is also a movement to put pressure on those countries and cultures, much like the pressure against apartheid in the 1980s and early 90s.