"Progressive" is the range of politics between the center-left and far left. For us Canadians it'd definitely be the NDP, and possibly even further to the left with things like the Courage Coalition (
http://www.couragecoalition.ca/), and for the US it'd be Democratic Socialists.
The term "movement" is meant to establish that it's operating at the level of individual citizens, not at the level of political party officials, think tanks and lobbyists, media, corporations, etc. — though often those latter groups will call what they're doing a "movement" in an attempt to seem more authentic and of the people.
The Sunrise Movement is probably a good example of a progressive movement focused on climate change:
https://en.wikipedia.org/wiki/Sunrise_Movement