They would cap carbon emissions, auction off carbon permits and redistribute the auction money to citizens. Cap and trade will make carbon intensive products and services more expensive, but the dividend means that low-carbon users would come out ahead.
Cap and dividend is a simple, market-based way to reduce CO2 emissions without reducing household income. It caps fossil fuel supplies, makes polluters pay, and returns the revenue to everyone equally.
It is modeled after the Alaska Permanent Fund, which pays equal dividends from the proceeds of state oil leases.
Having lived in Alaska myself, I can attest to the Permanent Fund's political popularity. No politician can touch it, and giving the money back to the citizens would more than address the denialists and their newfound concern for a consumption tax that affects the poor more than the rich. I think the left can be as simplistic as the right when it comes to consumption taxes - there are ways to keep them from being regressive.
The one problem with this idea is that if you don't use the auction proceeds to help control future emissions, which is what Obama wants to do with funding renewable energy research, then you have to set stricter standards to make up for the lost investment. But if it makes the whole thing more politically palatable, that's fine with me.