Just about anything by British director Ken Loach would fit the bill, including:
Land and Freedom
Riff-Raff
Ladybird, Ladybird
Navigators
and a bunch of his other stuff.
John Sayles is sort of an American equivalent, with plenty of his films dealing, either directly or obliquely, with social justice themes. Jeff Lichtman has already mentioned Brother from Another Planet, but you could check out some of Sayles’ other work, especially:
Matewan
City of Hope
Lone Star
Sunshine State
Go For Sister
The great thing about Loach and Sayles is that they’re definitely not directors who produce narratives of heroes and white knights. A lot of their stuff is about the value of collective action and working-class solidarity, and because they’re realists who often base their movies on actual historical events, the “good guys” often end up losing.
Another movie that might fit the bill is the New Zealand film Once Were Warriors, about working-class Maori communities in Auckland. It’s pretty hard to watch in places, but i think it’s an excellent movie.
For race in the United States, Crash is an obvious contender, but i thought it was a terrible movie. Some of Spike Lee’s stuff like Do the Right Thing (already mentioned above) and even Bamboozled, which doesn’t get much love, is worth checking out.
For gay rights and homosexuality, you could consider Milk, And the Band Played On, and Brokeback Mountain.